Results of Bernoulli and Euler Polynomials

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
24.2.E1 t e t - 1 = βˆ‘ n = 0 ∞ B n ⁒ t n n ! 𝑑 superscript 𝑒 𝑑 1 superscript subscript 𝑛 0 Bernoulli-number-B 𝑛 superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^% {n}}{n!}}} (t)/(exp(t)- 1)= sum(bernoulli(n)*((t)^(n))/(factorial(n)), n = 0..infinity) Divide[t,Exp[t]- 1]= Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] Failure Successful Skip -
24.2#Ex1 B 2 ⁒ n + 1 = 0 Bernoulli-number-B 2 𝑛 1 0 {\displaystyle{\displaystyle B_{2n+1}=0}} bernoulli(2*n + 1)= 0 BernoulliB[2*n + 1]= 0 Failure Failure Successful
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[BernoulliB[Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
24.2#Ex2 ( - 1 ) n + 1 ⁒ B 2 ⁒ n > 0 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 0 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>0}} (- 1)^(n + 1)* bernoulli(2*n)> 0 (- 1)^(n + 1)* BernoulliB[2*n]> 0 Failure Failure Successful Successful
24.2.E3 t ⁒ e x ⁒ t e t - 1 = βˆ‘ n = 0 ∞ B n ⁑ ( x ) ⁒ t n n ! 𝑑 superscript 𝑒 π‘₯ 𝑑 superscript 𝑒 𝑑 1 superscript subscript 𝑛 0 Bernoulli-polynomial-B 𝑛 π‘₯ superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}% \left(x\right)\frac{t^{n}}{n!}}} (t*exp(x*t))/(exp(t)- 1)= sum(bernoulli(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity) Divide[t*Exp[x*t],Exp[t]- 1]= Sum[BernoulliB[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] Failure Successful Skip -
24.2.E4 B n = B n ⁑ ( 0 ) Bernoulli-number-B 𝑛 Bernoulli-polynomial-B 𝑛 0 {\displaystyle{\displaystyle B_{n}=B_{n}\left(0\right)}} bernoulli(n)= bernoulli(n, 0) BernoulliB[n]= BernoulliB[n, 0] Successful Successful - -
24.2.E6 2 ⁒ e t e 2 ⁒ t + 1 = βˆ‘ n = 0 ∞ E n ⁒ t n n ! 2 superscript 𝑒 𝑑 superscript 𝑒 2 𝑑 1 superscript subscript 𝑛 0 Euler-number-E 𝑛 superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle\frac{2e^{t}}{e^{2t}+1}=\sum_{n=0}^{\infty}E_{n}% \frac{t^{n}}{n!}}} Error Divide[2*Exp[t],Exp[2*t]+ 1]= Sum[EulerE[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] Error Successful - -
24.2#Ex3 E 2 ⁒ n + 1 = 0 Euler-number-E 2 𝑛 1 0 {\displaystyle{\displaystyle E_{2n+1}=0}} Error EulerE[2*n + 1]= 0 Error Failure -
Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[EulerE[Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
24.2#Ex4 ( - 1 ) n ⁒ E 2 ⁒ n > 0 superscript 1 𝑛 Euler-number-E 2 𝑛 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>0}} Error (- 1)^(n)* EulerE[2*n]> 0 Error Failure - Successful
24.2.E8 2 ⁒ e x ⁒ t e t + 1 = βˆ‘ n = 0 ∞ E n ⁑ ( x ) ⁒ t n n ! 2 superscript 𝑒 π‘₯ 𝑑 superscript 𝑒 𝑑 1 superscript subscript 𝑛 0 Euler-polynomial-E 𝑛 π‘₯ superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}% \left(x\right)\frac{t^{n}}{n!}}} (2*exp(x*t))/(exp(t)+ 1)= sum(euler(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity) Divide[2*Exp[x*t],Exp[t]+ 1]= Sum[EulerE[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] Failure Successful Skip -
24.2.E9 E n = 2 n ⁒ E n ⁑ ( 1 2 ) Euler-number-E 𝑛 superscript 2 𝑛 Euler-polynomial-E 𝑛 1 2 {\displaystyle{\displaystyle E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)}} Error EulerE[n]= (2)^(n)* EulerE[n, Divide[1,2]] Error Successful - -
24.4.E1 B n ⁑ ( x + 1 ) - B n ⁑ ( x ) = n ⁒ x n - 1 Bernoulli-polynomial-B 𝑛 π‘₯ 1 Bernoulli-polynomial-B 𝑛 π‘₯ 𝑛 superscript π‘₯ 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(x+1\right)-B_{n}\left(x\right)=nx^{n-1% }}} bernoulli(n, x + 1)- bernoulli(n, x)= n*(x)^(n - 1) BernoulliB[n, x + 1]- BernoulliB[n, x]= n*(x)^(n - 1) Successful Successful - -
24.4.E2 E n ⁑ ( x + 1 ) + E n ⁑ ( x ) = 2 ⁒ x n Euler-polynomial-E 𝑛 π‘₯ 1 Euler-polynomial-E 𝑛 π‘₯ 2 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle E_{n}\left(x+1\right)+E_{n}\left(x\right)=2x^{n}}} euler(n, x + 1)+ euler(n, x)= 2*(x)^(n) EulerE[n, x + 1]+ EulerE[n, x]= 2*(x)^(n) Successful Successful - -
24.4.E3 B n ⁑ ( 1 - x ) = ( - 1 ) n ⁒ B n ⁑ ( x ) Bernoulli-polynomial-B 𝑛 1 π‘₯ superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 π‘₯ {\displaystyle{\displaystyle B_{n}\left(1-x\right)=(-1)^{n}B_{n}\left(x\right)}} bernoulli(n, 1 - x)=(- 1)^(n)* bernoulli(n, x) BernoulliB[n, 1 - x]=(- 1)^(n)* BernoulliB[n, x] Successful Failure - Successful
24.4.E4 E n ⁑ ( 1 - x ) = ( - 1 ) n ⁒ E n ⁑ ( x ) Euler-polynomial-E 𝑛 1 π‘₯ superscript 1 𝑛 Euler-polynomial-E 𝑛 π‘₯ {\displaystyle{\displaystyle E_{n}\left(1-x\right)=(-1)^{n}E_{n}\left(x\right)}} euler(n, 1 - x)=(- 1)^(n)* euler(n, x) EulerE[n, 1 - x]=(- 1)^(n)* EulerE[n, x] Successful Failure - Successful
24.4.E5 ( - 1 ) n ⁒ B n ⁑ ( - x ) = B n ⁑ ( x ) + n ⁒ x n - 1 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 π‘₯ Bernoulli-polynomial-B 𝑛 π‘₯ 𝑛 superscript π‘₯ 𝑛 1 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(-x\right)=B_{n}\left(x\right)+% nx^{n-1}}} (- 1)^(n)* bernoulli(n, - x)= bernoulli(n, x)+ n*(x)^(n - 1) (- 1)^(n)* BernoulliB[n, - x]= BernoulliB[n, x]+ n*(x)^(n - 1) Failure Failure Successful Successful
24.4.E6 ( - 1 ) n + 1 ⁒ E n ⁑ ( - x ) = E n ⁑ ( x ) - 2 ⁒ x n superscript 1 𝑛 1 Euler-polynomial-E 𝑛 π‘₯ Euler-polynomial-E 𝑛 π‘₯ 2 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle(-1)^{n+1}E_{n}\left(-x\right)=E_{n}\left(x\right)% -2x^{n}}} (- 1)^(n + 1)* euler(n, - x)= euler(n, x)- 2*(x)^(n) (- 1)^(n + 1)* EulerE[n, - x]= EulerE[n, x]- 2*(x)^(n) Failure Failure Successful Successful
24.4.E7 βˆ‘ k = 1 m k n = B n + 1 ⁑ ( m + 1 ) - B n + 1 n + 1 superscript subscript π‘˜ 1 π‘š superscript π‘˜ 𝑛 Bernoulli-polynomial-B 𝑛 1 π‘š 1 Bernoulli-number-B 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\sum_{k=1}^{m}k^{n}=\frac{B_{n+1}\left(m+1\right)-% B_{n+1}}{n+1}}} sum((k)^(n), k = 1..m)=(bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1) Sum[(k)^(n), {k, 1, m}]=Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1] Failure Failure Skip Successful
24.4.E8 βˆ‘ k = 1 m ( - 1 ) m - k ⁒ k n = E n ⁑ ( m + 1 ) + ( - 1 ) m ⁒ E n ⁑ ( 0 ) 2 superscript subscript π‘˜ 1 π‘š superscript 1 π‘š π‘˜ superscript π‘˜ 𝑛 Euler-polynomial-E 𝑛 π‘š 1 superscript 1 π‘š Euler-polynomial-E 𝑛 0 2 {\displaystyle{\displaystyle\sum_{k=1}^{m}(-1)^{m-k}k^{n}=\frac{E_{n}\left(m+1% \right)+(-1)^{m}E_{n}\left(0\right)}{2}}} sum((- 1)^(m - k)* (k)^(n), k = 1..m)=(euler(n, m + 1)+(- 1)^(m)* euler(n, 0))/(2) Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}]=Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2] Failure Failure Skip Successful
24.4.E9 βˆ‘ k = 0 m - 1 ( a + d ⁒ k ) n = d n n + 1 ⁒ ( B n + 1 ⁑ ( m + a d ) - B n + 1 ⁑ ( a d ) ) superscript subscript π‘˜ 0 π‘š 1 superscript π‘Ž 𝑑 π‘˜ 𝑛 superscript 𝑑 𝑛 𝑛 1 Bernoulli-polynomial-B 𝑛 1 π‘š π‘Ž 𝑑 Bernoulli-polynomial-B 𝑛 1 π‘Ž 𝑑 {\displaystyle{\displaystyle\sum_{k=0}^{m-1}(a+dk)^{n}={\frac{d^{n}}{n+1}\left% (B_{n+1}\left(m+\frac{a}{d}\right)-B_{n+1}\left(\frac{a}{d}\right)\right)}}} sum((a + d*k)^(n), k = 0..m - 1)=((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d))) Sum[(a + d*k)^(n), {k, 0, m - 1}]=Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]]) Failure Failure Skip Skip
24.4.E10 βˆ‘ k = 0 m - 1 ( - 1 ) k ⁒ ( a + d ⁒ k ) n = d n 2 ⁒ ( ( - 1 ) m - 1 ⁒ E n ⁑ ( m + a d ) + E n ⁑ ( a d ) ) superscript subscript π‘˜ 0 π‘š 1 superscript 1 π‘˜ superscript π‘Ž 𝑑 π‘˜ 𝑛 superscript 𝑑 𝑛 2 superscript 1 π‘š 1 Euler-polynomial-E 𝑛 π‘š π‘Ž 𝑑 Euler-polynomial-E 𝑛 π‘Ž 𝑑 {\displaystyle{\displaystyle\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n}={\frac{d^{n}}{2% }\left((-1)^{m-1}E_{n}\left(m+\frac{a}{d}\right)+E_{n}\left(\frac{a}{d}\right)% \right)}}} sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1)=((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d))) Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}]=Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]]) Failure Failure Skip Skip
24.4.E21 B n ⁑ ( x ) = 2 n - 1 ⁒ ( B n ⁑ ( 1 2 ⁒ x ) + B n ⁑ ( 1 2 ⁒ x + 1 2 ) ) Bernoulli-polynomial-B 𝑛 π‘₯ superscript 2 𝑛 1 Bernoulli-polynomial-B 𝑛 1 2 π‘₯ Bernoulli-polynomial-B 𝑛 1 2 π‘₯ 1 2 {\displaystyle{\displaystyle B_{n}\left(x\right)=2^{n-1}\left(B_{n}\left(% \tfrac{1}{2}x\right)+B_{n}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)\right)}} bernoulli(n, x)= (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2))) BernoulliB[n, x]= (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]]) Failure Failure Successful Successful
24.4.E22 E n - 1 ⁑ ( x ) = 2 n ⁒ ( B n ⁑ ( x ) - 2 n ⁒ B n ⁑ ( 1 2 ⁒ x ) ) Euler-polynomial-E 𝑛 1 π‘₯ 2 𝑛 Bernoulli-polynomial-B 𝑛 π‘₯ superscript 2 𝑛 Bernoulli-polynomial-B 𝑛 1 2 π‘₯ {\displaystyle{\displaystyle E_{n-1}\left(x\right)=\frac{2}{n}\left(B_{n}\left% (x\right)-2^{n}B_{n}\left(\tfrac{1}{2}x\right)\right)}} euler(n - 1, x)=(2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x)) EulerE[n - 1, x]=Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x]) Failure Failure Successful Successful
24.4.E23 E n - 1 ⁑ ( x ) = 2 n n ⁒ ( B n ⁑ ( 1 2 ⁒ x + 1 2 ) - B n ⁑ ( 1 2 ⁒ x ) ) Euler-polynomial-E 𝑛 1 π‘₯ superscript 2 𝑛 𝑛 Bernoulli-polynomial-B 𝑛 1 2 π‘₯ 1 2 Bernoulli-polynomial-B 𝑛 1 2 π‘₯ {\displaystyle{\displaystyle E_{n-1}\left(x\right)=\frac{2^{n}}{n}\left(B_{n}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)-B_{n}\left(\tfrac{1}{2}x\right)\right)}} euler(n - 1, x)=((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x)) EulerE[n - 1, x]=Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x]) Failure Failure Successful Successful
24.4.E25 B n ⁑ ( 0 ) = ( - 1 ) n ⁒ B n ⁑ ( 1 ) Bernoulli-polynomial-B 𝑛 0 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(0\right)=(-1)^{n}B_{n}\left(1\right)}} bernoulli(n, 0)=(- 1)^(n)* bernoulli(n, 1) BernoulliB[n, 0]=(- 1)^(n)* BernoulliB[n, 1] Failure Failure Successful Successful
24.4.E25 ( - 1 ) n ⁒ B n ⁑ ( 1 ) = B n superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 1 Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(1\right)=B_{n}}} (- 1)^(n)* bernoulli(n, 1)= bernoulli(n) (- 1)^(n)* BernoulliB[n, 1]= BernoulliB[n] Failure Failure Successful Successful
24.4.E26 E n ⁑ ( 0 ) = - E n ⁑ ( 1 ) Euler-polynomial-E 𝑛 0 Euler-polynomial-E 𝑛 1 {\displaystyle{\displaystyle E_{n}\left(0\right)=-E_{n}\left(1\right)}} euler(n, 0)= - euler(n, 1) EulerE[n, 0]= - EulerE[n, 1] Successful Successful - -
24.4.E26 - E n ⁑ ( 1 ) = - 2 n + 1 ⁒ ( 2 n + 1 - 1 ) ⁒ B n + 1 Euler-polynomial-E 𝑛 1 2 𝑛 1 superscript 2 𝑛 1 1 Bernoulli-number-B 𝑛 1 {\displaystyle{\displaystyle-E_{n}\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n% +1}}} - euler(n, 1)= -(2)/(n + 1)*((2)^(n + 1)- 1)* bernoulli(n + 1) - EulerE[n, 1]= -Divide[2,n + 1]*((2)^(n + 1)- 1)* BernoulliB[n + 1] Failure Failure Error Successful
24.4.E27 B n ⁑ ( 1 2 ) = - ( 1 - 2 1 - n ) ⁒ B n Bernoulli-polynomial-B 𝑛 1 2 1 superscript 2 1 𝑛 Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{n}\left(\tfrac{1}{2}\right)=-(1-2^{1-n})B_{n}}} bernoulli(n, (1)/(2))= -(1 - (2)^(1 - n))* bernoulli(n) BernoulliB[n, Divide[1,2]]= -(1 - (2)^(1 - n))* BernoulliB[n] Successful Successful - -
24.4.E28 E n ⁑ ( 1 2 ) = 2 - n ⁒ E n Euler-polynomial-E 𝑛 1 2 superscript 2 𝑛 Euler-number-E 𝑛 {\displaystyle{\displaystyle E_{n}\left(\tfrac{1}{2}\right)=2^{-n}E_{n}}} Error EulerE[n, Divide[1,2]]= (2)^(- n)* EulerE[n] Error Successful - -
24.4.E29 B 2 ⁒ n ⁑ ( 1 3 ) = B 2 ⁒ n ⁑ ( 2 3 ) Bernoulli-polynomial-B 2 𝑛 1 3 Bernoulli-polynomial-B 2 𝑛 2 3 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{1}{3}\right)=B_{2n}\left(% \tfrac{2}{3}\right)}} bernoulli(2*n, (1)/(3))= bernoulli(2*n, (2)/(3)) BernoulliB[2*n, Divide[1,3]]= BernoulliB[2*n, Divide[2,3]] Failure Failure Successful
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.4.E29 B 2 ⁒ n ⁑ ( 2 3 ) = - 1 2 ⁒ ( 1 - 3 1 - 2 ⁒ n ) ⁒ B 2 ⁒ n Bernoulli-polynomial-B 2 𝑛 2 3 1 2 1 superscript 3 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{2}{3}\right)=-\tfrac{1}{2}(1-3% ^{1-2n})B_{2n}}} bernoulli(2*n, (2)/(3))= -(1)/(2)*(1 - (3)^(1 - 2*n))* bernoulli(2*n) BernoulliB[2*n, Divide[2,3]]= -Divide[1,2]*(1 - (3)^(1 - 2*n))* BernoulliB[2*n] Failure Failure Successful Successful
24.4.E30 E 2 ⁒ n - 1 ⁑ ( 1 3 ) = - E 2 ⁒ n - 1 ⁑ ( 2 3 ) Euler-polynomial-E 2 𝑛 1 1 3 Euler-polynomial-E 2 𝑛 1 2 3 {\displaystyle{\displaystyle E_{2n-1}\left(\tfrac{1}{3}\right)=-E_{2n-1}\left(% \tfrac{2}{3}\right)}} euler(2*n - 1, (1)/(3))= - euler(2*n - 1, (2)/(3)) EulerE[2*n - 1, Divide[1,3]]= - EulerE[2*n - 1, Divide[2,3]] Failure Failure Successful
Fail
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 2.8284271247461903] <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[EulerE[Plus[-1, Times[2, n]], Rational[1, 3]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[EulerE[Plus[-1, Times[2, n]], Rational[2, 3]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.4.E30 - E 2 ⁒ n - 1 ⁑ ( 2 3 ) = - ( 1 - 3 1 - 2 ⁒ n ) ⁒ ( 2 2 ⁒ n - 1 ) 2 ⁒ n ⁒ B 2 ⁒ n Euler-polynomial-E 2 𝑛 1 2 3 1 superscript 3 1 2 𝑛 superscript 2 2 𝑛 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle-E_{2n-1}\left(\tfrac{2}{3}\right)=-\frac{(1-3^{1-% 2n})(2^{2n}-1)}{2n}B_{2n}}} - euler(2*n - 1, (2)/(3))= -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n) - EulerE[2*n - 1, Divide[2,3]]= -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n] Failure Failure Successful Successful
24.4.E31 B n ⁑ ( 1 4 ) = ( - 1 ) n ⁒ B n ⁑ ( 3 4 ) Bernoulli-polynomial-B 𝑛 1 4 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 3 4 {\displaystyle{\displaystyle B_{n}\left(\tfrac{1}{4}\right)=(-1)^{n}B_{n}\left% (\tfrac{3}{4}\right)}} bernoulli(n, (1)/(4))=(- 1)^(n)* bernoulli(n, (3)/(4)) BernoulliB[n, Divide[1,4]]=(- 1)^(n)* BernoulliB[n, Divide[3,4]] Failure Successful Successful -
24.4.E31 ( - 1 ) n ⁒ B n ⁑ ( 3 4 ) = - 1 - 2 1 - n 2 n ⁒ B n - n 4 n ⁒ E n - 1 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 3 4 1 superscript 2 1 𝑛 superscript 2 𝑛 Bernoulli-number-B 𝑛 𝑛 superscript 4 𝑛 Euler-number-E 𝑛 1 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(\tfrac{3}{4}\right)=-\frac{1-2^% {1-n}}{2^{n}}B_{n}-\frac{n}{4^{n}}E_{n-1}}} Error (- 1)^(n)* BernoulliB[n, Divide[3,4]]= -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1] Error Failure - Successful
24.4.E32 B 2 ⁒ n ⁑ ( 1 6 ) = B 2 ⁒ n ⁑ ( 5 6 ) Bernoulli-polynomial-B 2 𝑛 1 6 Bernoulli-polynomial-B 2 𝑛 5 6 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{1}{6}\right)=B_{2n}\left(% \tfrac{5}{6}\right)}} bernoulli(2*n, (1)/(6))= bernoulli(2*n, (5)/(6)) BernoulliB[2*n, Divide[1,6]]= BernoulliB[2*n, Divide[5,6]] Failure Failure Successful
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[Times[2, n], Rational[1, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[BernoulliB[Times[2, n], Rational[5, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.4.E32 B 2 ⁒ n ⁑ ( 5 6 ) = 1 2 ⁒ ( 1 - 2 1 - 2 ⁒ n ) ⁒ ( 1 - 3 1 - 2 ⁒ n ) ⁒ B 2 ⁒ n Bernoulli-polynomial-B 2 𝑛 5 6 1 2 1 superscript 2 1 2 𝑛 1 superscript 3 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{5}{6}\right)=\tfrac{1}{2}(1-2^% {1-2n})(1-3^{1-2n})B_{2n}}} bernoulli(2*n, (5)/(6))=(1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* bernoulli(2*n) BernoulliB[2*n, Divide[5,6]]=Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))* BernoulliB[2*n] Failure Failure Successful Successful
24.4.E33 E 2 ⁒ n ⁑ ( 1 6 ) = E 2 ⁒ n ⁑ ( 5 6 ) Euler-polynomial-E 2 𝑛 1 6 Euler-polynomial-E 2 𝑛 5 6 {\displaystyle{\displaystyle E_{2n}\left(\tfrac{1}{6}\right)=E_{2n}\left(% \tfrac{5}{6}\right)}} euler(2*n, (1)/(6))= euler(2*n, (5)/(6)) EulerE[2*n, Divide[1,6]]= EulerE[2*n, Divide[5,6]] Failure Failure Successful
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[EulerE[Times[2, n], Rational[1, 6]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[EulerE[Times[2, n], Rational[5, 6]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.4.E33 E 2 ⁒ n ⁑ ( 5 6 ) = 1 + 3 - 2 ⁒ n 2 2 ⁒ n + 1 ⁒ E 2 ⁒ n Euler-polynomial-E 2 𝑛 5 6 1 superscript 3 2 𝑛 superscript 2 2 𝑛 1 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle E_{2n}\left(\tfrac{5}{6}\right)=\frac{1+3^{-2n}}{% 2^{2n+1}}E_{2n}}} Error EulerE[2*n, Divide[5,6]]=Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n] Error Failure - Successful
24.4.E34 d d x ⁑ B n ⁑ ( x ) = n ⁒ B n - 1 ⁑ ( x ) derivative π‘₯ Bernoulli-polynomial-B 𝑛 π‘₯ 𝑛 Bernoulli-polynomial-B 𝑛 1 π‘₯ {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}B_{n}\left(x\right)=% nB_{n-1}\left(x\right)}} diff(bernoulli(n, x), x)= n*bernoulli(n - 1, x) D[BernoulliB[n, x], x]= n*BernoulliB[n - 1, x] Successful Successful - -
24.4.E35 d d x ⁑ E n ⁑ ( x ) = n ⁒ E n - 1 ⁑ ( x ) derivative π‘₯ Euler-polynomial-E 𝑛 π‘₯ 𝑛 Euler-polynomial-E 𝑛 1 π‘₯ {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}E_{n}\left(x\right)=% nE_{n-1}\left(x\right)}} diff(euler(n, x), x)= n*euler(n - 1, x) D[EulerE[n, x], x]= n*EulerE[n - 1, x] Successful Successful - -
24.4.E37 B n ⁑ ( x + h ) = ( B ⁒ ( x ) + h ) n Bernoulli-polynomial-B 𝑛 π‘₯ β„Ž superscript 𝐡 π‘₯ β„Ž 𝑛 {\displaystyle{\displaystyle B_{n}\left(x+h\right)=(B(x)+h)^{n}}} bernoulli(n, x + h)=(B*(x)+ h)^(n) BernoulliB[n, x + h]=(B*(x)+ h)^(n) Failure Failure
Fail
-.9142135620-1.414213562*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
-1.328427124-2.828427124*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-1.742640686-4.242640686*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
1.580880229-10.58578643*I <- {B = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Fail
Complex[-0.9142135623730951, -1.4142135623730951] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
Complex[-1.3284271247461903, -2.8284271247461903] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
Complex[-1.7426406871192857, -4.242640687119286] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[1.580880229039761, -10.585786437626906] <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
... skip entries to safe data
24.4.E39 E n ⁑ ( x + h ) = ( E ⁒ ( x ) + h ) n Euler-polynomial-E 𝑛 π‘₯ β„Ž superscript 𝐸 π‘₯ β„Ž 𝑛 {\displaystyle{\displaystyle E_{n}\left(x+h\right)=(E(x)+h)^{n}}} euler(n, x + h)=(E*(x)+ h)^(n) EulerE[n, x + h]=(E*(x)+ h)^(n) Failure Failure
Fail
-.9142135620-1.414213562*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
-1.328427124-2.828427124*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-1.742640686-4.242640686*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
1.414213562-10.58578643*I <- {E = 2^(1/2)+I*2^(1/2), h = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Fail
-2.218281828459045 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1]}
-3.93656365691809 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 2]}
-5.654845485377136 <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 3]}
Complex[-13.66330459287579, -6.27424849394514] <- {Rule[h, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1]}
... skip entries to safe data
24.5.E8 βˆ‘ k = 0 n 2 2 ⁒ k ⁒ B 2 ⁒ k ( 2 ⁒ k ) ! ⁒ ( 2 ⁒ n + 1 - 2 ⁒ k ) ! = 1 ( 2 ⁒ n ) ! superscript subscript π‘˜ 0 𝑛 superscript 2 2 π‘˜ Bernoulli-number-B 2 π‘˜ 2 π‘˜ 2 𝑛 1 2 π‘˜ 1 2 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n}\frac{2^{2k}B_{2k}}{(2k)!(2n+1-2k)!}% =\frac{1}{(2n)!}}} sum(((2)^(2*k)* bernoulli(2*k))/(factorial(2*k)*factorial(2*n + 1 - 2*k)), k = 0..n)=(1)/(factorial(2*n)) Sum[Divide[(2)^(2*k)* BernoulliB[2*k],(2*k)!*(2*n + 1 - 2*k)!], {k, 0, n}]=Divide[1,(2*n)!] Failure Failure Skip Successful
24.6.E2 B n = 1 n + 1 ⁒ βˆ‘ k = 1 n βˆ‘ j = 1 k ( - 1 ) j ⁒ j n ⁒ ( n + 1 k - j ) / ( n k ) Bernoulli-number-B 𝑛 1 𝑛 1 superscript subscript π‘˜ 1 𝑛 superscript subscript 𝑗 1 π‘˜ superscript 1 𝑗 superscript 𝑗 𝑛 binomial 𝑛 1 π‘˜ 𝑗 binomial 𝑛 π‘˜ {\displaystyle{\displaystyle B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-% 1)^{j}j^{n}{\genfrac{(}{)}{0.0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}% {n}{k}}}} bernoulli(n)=(1)/(n + 1)*sum(sum((- 1)^(j)* (j)^(n)*binomial(n + 1,k - j)/binomial(n,k), j = 1..k), k = 1..n) BernoulliB[n]=Divide[1,n + 1]*Sum[Sum[(- 1)^(j)* (j)^(n)*Binomial[n + 1,k - j]/Binomial[n,k], {j, 1, k}], {k, 1, n}] Failure Failure Skip Successful
24.7.E1 B 2 ⁒ n = ( - 1 ) n + 1 ⁒ 4 ⁒ n 1 - 2 1 - 2 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 e 2 ⁒ Ο€ ⁒ t + 1 ⁒ d t Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 4 𝑛 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 2 πœ‹ 𝑑 1 𝑑 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{% \infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\mathrm{d}t}} bernoulli(2*n)=(- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity) BernoulliB[2*n]=(- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}] Failure Failure Skip Successful
24.7.E1 ( - 1 ) n + 1 ⁒ 4 ⁒ n 1 - 2 1 - 2 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 e 2 ⁒ Ο€ ⁒ t + 1 ⁒ d t = ( - 1 ) n + 1 ⁒ 2 ⁒ n 1 - 2 1 - 2 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 ⁒ e - Ο€ ⁒ t ⁒ sech ⁑ ( Ο€ ⁒ t ) ⁒ d t superscript 1 𝑛 1 4 𝑛 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 2 πœ‹ 𝑑 1 𝑑 superscript 1 𝑛 1 2 𝑛 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 πœ‹ 𝑑 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}% \frac{t^{2n-1}}{e^{2\pi t}+1}\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{% 0}^{\infty}t^{2n-1}e^{-\pi t}\operatorname{sech}\left(\pi t\right)\mathrm{d}t}} (- 1)^(n + 1)*(4*n)/(1 - (2)^(1 - 2*n))*int(((t)^(2*n - 1))/(exp(2*Pi*t)+ 1), t = 0..infinity)=(- 1)^(n + 1)*(2*n)/(1 - (2)^(1 - 2*n))*int((t)^(2*n - 1)* exp(- Pi*t)*sech(Pi*t), t = 0..infinity) (- 1)^(n + 1)*Divide[4*n,1 - (2)^(1 - 2*n)]*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]+ 1], {t, 0, Infinity}]=(- 1)^(n + 1)*Divide[2*n,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Sech[Pi*t], {t, 0, Infinity}] Successful Failure - Skip
24.7.E2 B 2 ⁒ n = ( - 1 ) n + 1 ⁒ 4 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 e 2 ⁒ Ο€ ⁒ t - 1 ⁒ d t Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 4 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 2 πœ‹ 𝑑 1 𝑑 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1% }}{e^{2\pi t}-1}\mathrm{d}t}} bernoulli(2*n)=(- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity) BernoulliB[2*n]=(- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}] Failure Failure Skip Successful
24.7.E2 ( - 1 ) n + 1 ⁒ 4 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 e 2 ⁒ Ο€ ⁒ t - 1 ⁒ d t = ( - 1 ) n + 1 ⁒ 2 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n - 1 ⁒ e - Ο€ ⁒ t ⁒ csch ⁑ ( Ο€ ⁒ t ) ⁒ d t superscript 1 𝑛 1 4 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 2 πœ‹ 𝑑 1 𝑑 superscript 1 𝑛 1 2 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 1 superscript 𝑒 πœ‹ 𝑑 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle(-1)^{n+1}4n\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2% \pi t}-1}\mathrm{d}t=(-1)^{n+1}2n\int_{0}^{\infty}t^{2n-1}e^{-\pi t}% \operatorname{csch}\left(\pi t\right)\mathrm{d}t}} (- 1)^(n + 1)* 4*n*int(((t)^(2*n - 1))/(exp(2*Pi*t)- 1), t = 0..infinity)=(- 1)^(n + 1)* 2*n*int((t)^(2*n - 1)* exp(- Pi*t)*csch(Pi*t), t = 0..infinity) (- 1)^(n + 1)* 4*n*Integrate[Divide[(t)^(2*n - 1),Exp[2*Pi*t]- 1], {t, 0, Infinity}]=(- 1)^(n + 1)* 2*n*Integrate[(t)^(2*n - 1)* Exp[- Pi*t]*Csch[Pi*t], {t, 0, Infinity}] Successful Failure - Skip
24.7.E3 B 2 ⁒ n = ( - 1 ) n + 1 ⁒ Ο€ 1 - 2 1 - 2 ⁒ n ⁒ ∫ 0 ∞ t 2 ⁒ n ⁒ sech 2 ⁑ ( Ο€ ⁒ t ) ⁒ d t Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 πœ‹ 1 superscript 2 1 2 𝑛 superscript subscript 0 superscript 𝑑 2 𝑛 2 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}\frac{\pi}{1-2^{1-2n}}\int_{0}^{% \infty}t^{2n}{\operatorname{sech}^{2}}\left(\pi t\right)\mathrm{d}t}} bernoulli(2*n)=(- 1)^(n + 1)*(Pi)/(1 - (2)^(1 - 2*n))*int((t)^(2*n)* (sech(Pi*t))^(2), t = 0..infinity) BernoulliB[2*n]=(- 1)^(n + 1)*Divide[Pi,1 - (2)^(1 - 2*n)]*Integrate[(t)^(2*n)* (Sech[Pi*t])^(2), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E4 B 2 ⁒ n = ( - 1 ) n + 1 ⁒ Ο€ ⁒ ∫ 0 ∞ t 2 ⁒ n ⁒ csch 2 ⁑ ( Ο€ ⁒ t ) ⁒ d t Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 πœ‹ superscript subscript 0 superscript 𝑑 2 𝑛 2 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}\pi\int_{0}^{\infty}t^{2n}{% \operatorname{csch}^{2}}\left(\pi t\right)\mathrm{d}t}} bernoulli(2*n)=(- 1)^(n + 1)* Pi*int((t)^(2*n)* (csch(Pi*t))^(2), t = 0..infinity) BernoulliB[2*n]=(- 1)^(n + 1)* Pi*Integrate[(t)^(2*n)* (Csch[Pi*t])^(2), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E5 B 2 ⁒ n = ( - 1 ) n ⁒ 2 ⁒ n ⁒ ( 2 ⁒ n - 1 ) Ο€ ⁒ ∫ 0 ∞ t 2 ⁒ n - 2 ⁒ ln ⁑ ( 1 - e - 2 ⁒ Ο€ ⁒ t ) ⁒ d t Bernoulli-number-B 2 𝑛 superscript 1 𝑛 2 𝑛 2 𝑛 1 πœ‹ superscript subscript 0 superscript 𝑑 2 𝑛 2 1 superscript 𝑒 2 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle B_{2n}=(-1)^{n}\frac{2n(2n-1)}{\pi}\*\int_{0}^{% \infty}t^{2n-2}\ln\left(1-e^{-2\pi t}\right)\mathrm{d}t}} bernoulli(2*n)=(- 1)^(n)*(2*n*(2*n - 1))/(Pi)* int((t)^(2*n - 2)* ln(1 - exp(- 2*Pi*t)), t = 0..infinity) BernoulliB[2*n]=(- 1)^(n)*Divide[2*n*(2*n - 1),Pi]* Integrate[(t)^(2*n - 2)* Log[1 - Exp[- 2*Pi*t]], {t, 0, Infinity}] Failure Failure Skip Successful
24.7.E6 E 2 ⁒ n = ( - 1 ) n ⁒ 2 2 ⁒ n + 1 ⁒ ∫ 0 ∞ t 2 ⁒ n ⁒ sech ⁑ ( Ο€ ⁒ t ) ⁒ d t Euler-number-E 2 𝑛 superscript 1 𝑛 superscript 2 2 𝑛 1 superscript subscript 0 superscript 𝑑 2 𝑛 πœ‹ 𝑑 𝑑 {\displaystyle{\displaystyle E_{2n}=(-1)^{n}2^{2n+1}\int_{0}^{\infty}t^{2n}% \operatorname{sech}\left(\pi t\right)\mathrm{d}t}} Error EulerE[2*n]=(- 1)^(n)* (2)^(2*n + 1)* Integrate[(t)^(2*n)* Sech[Pi*t], {t, 0, Infinity}] Error Failure - Skip
24.7.E7 B 2 ⁒ n ⁑ ( x ) = ( - 1 ) n + 1 ⁒ 2 ⁒ n ⁒ ∫ 0 ∞ cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) - e - 2 ⁒ Ο€ ⁒ t cosh ⁑ ( 2 ⁒ Ο€ ⁒ t ) - cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) ⁒ t 2 ⁒ n - 1 ⁒ d t Bernoulli-polynomial-B 2 𝑛 π‘₯ superscript 1 𝑛 1 2 𝑛 superscript subscript 0 2 πœ‹ π‘₯ superscript 𝑒 2 πœ‹ 𝑑 2 πœ‹ 𝑑 2 πœ‹ π‘₯ superscript 𝑑 2 𝑛 1 𝑑 {\displaystyle{\displaystyle B_{2n}\left(x\right)=(-1)^{n+1}2n\*\int_{0}^{% \infty}\frac{\cos\left(2\pi x\right)-e^{-2\pi t}}{\cosh\left(2\pi t\right)-% \cos\left(2\pi x\right)}t^{2n-1}\mathrm{d}t}} bernoulli(2*n, x)=(- 1)^(n + 1)* 2*n * int((cos(2*Pi*x)- exp(- 2*Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n - 1), t = 0..infinity) BernoulliB[2*n, x]=(- 1)^(n + 1)* 2*n * Integrate[Divide[Cos[2*Pi*x]- Exp[- 2*Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n - 1), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E8 B 2 ⁒ n + 1 ⁑ ( x ) = ( - 1 ) n + 1 ⁒ ( 2 ⁒ n + 1 ) ⁒ ∫ 0 ∞ sin ⁑ ( 2 ⁒ Ο€ ⁒ x ) cosh ⁑ ( 2 ⁒ Ο€ ⁒ t ) - cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) ⁒ t 2 ⁒ n ⁒ d t Bernoulli-polynomial-B 2 𝑛 1 π‘₯ superscript 1 𝑛 1 2 𝑛 1 superscript subscript 0 2 πœ‹ π‘₯ 2 πœ‹ 𝑑 2 πœ‹ π‘₯ superscript 𝑑 2 𝑛 𝑑 {\displaystyle{\displaystyle B_{2n+1}\left(x\right)=(-1)^{n+1}(2n+1)\*\int_{0}% ^{\infty}\frac{\sin\left(2\pi x\right)}{\cosh\left(2\pi t\right)-\cos\left(2% \pi x\right)}t^{2n}\mathrm{d}t}} bernoulli(2*n + 1, x)=(- 1)^(n + 1)*(2*n + 1)* int((sin(2*Pi*x))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity) BernoulliB[2*n + 1, x]=(- 1)^(n + 1)*(2*n + 1)* Integrate[Divide[Sin[2*Pi*x],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E9 E 2 ⁒ n ⁑ ( x ) = ( - 1 ) n ⁒ 4 ⁒ ∫ 0 ∞ sin ⁑ ( Ο€ ⁒ x ) ⁒ cosh ⁑ ( Ο€ ⁒ t ) cosh ⁑ ( 2 ⁒ Ο€ ⁒ t ) - cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) ⁒ t 2 ⁒ n ⁒ d t Euler-polynomial-E 2 𝑛 π‘₯ superscript 1 𝑛 4 superscript subscript 0 πœ‹ π‘₯ πœ‹ 𝑑 2 πœ‹ 𝑑 2 πœ‹ π‘₯ superscript 𝑑 2 𝑛 𝑑 {\displaystyle{\displaystyle E_{2n}\left(x\right)=(-1)^{n}4\int_{0}^{\infty}% \frac{\sin\left(\pi x\right)\cosh\left(\pi t\right)}{\cosh\left(2\pi t\right)-% \cos\left(2\pi x\right)}t^{2n}\mathrm{d}t}} euler(2*n, x)=(- 1)^(n)* 4*int((sin(Pi*x)*cosh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n), t = 0..infinity) EulerE[2*n, x]=(- 1)^(n)* 4*Integrate[Divide[Sin[Pi*x]*Cosh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E10 E 2 ⁒ n + 1 ⁑ ( x ) = ( - 1 ) n + 1 ⁒ 4 ⁒ ∫ 0 ∞ cos ⁑ ( Ο€ ⁒ x ) ⁒ sinh ⁑ ( Ο€ ⁒ t ) cosh ⁑ ( 2 ⁒ Ο€ ⁒ t ) - cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) ⁒ t 2 ⁒ n + 1 ⁒ d t Euler-polynomial-E 2 𝑛 1 π‘₯ superscript 1 𝑛 1 4 superscript subscript 0 πœ‹ π‘₯ πœ‹ 𝑑 2 πœ‹ 𝑑 2 πœ‹ π‘₯ superscript 𝑑 2 𝑛 1 𝑑 {\displaystyle{\displaystyle E_{2n+1}\left(x\right)=(-1)^{n+1}4\*\int_{0}^{% \infty}\frac{\cos\left(\pi x\right)\sinh\left(\pi t\right)}{\cosh\left(2\pi t% \right)-\cos\left(2\pi x\right)}t^{2n+1}\mathrm{d}t}} euler(2*n + 1, x)=(- 1)^(n + 1)* 4 * int((cos(Pi*x)*sinh(Pi*t))/(cosh(2*Pi*t)- cos(2*Pi*x))*(t)^(2*n + 1), t = 0..infinity) EulerE[2*n + 1, x]=(- 1)^(n + 1)* 4 * Integrate[Divide[Cos[Pi*x]*Sinh[Pi*t],Cosh[2*Pi*t]- Cos[2*Pi*x]]*(t)^(2*n + 1), {t, 0, Infinity}] Failure Failure Skip Error
24.7.E11 B n ⁑ ( x ) = 1 2 ⁒ Ο€ ⁒ i ⁒ ∫ - c - i ⁒ ∞ - c + i ⁒ ∞ ( x + t ) n ⁒ ( Ο€ sin ⁑ ( Ο€ ⁒ t ) ) 2 ⁒ d t Bernoulli-polynomial-B 𝑛 π‘₯ 1 2 πœ‹ 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript π‘₯ 𝑑 𝑛 superscript πœ‹ πœ‹ 𝑑 2 𝑑 {\displaystyle{\displaystyle B_{n}\left(x\right)=\frac{1}{2\pi i}\int_{-c-i% \infty}^{-c+i\infty}(x+t)^{n}\left(\frac{\pi}{\sin\left(\pi t\right)}\right)^{% 2}\mathrm{d}t}} bernoulli(n, x)=(1)/(2*Pi*I)*int((x + t)^(n)*((Pi)/(sin(Pi*t)))^(2), t = - c - I*infinity..- c + I*infinity) BernoulliB[n, x]=Divide[1,2*Pi*I]*Integrate[(x + t)^(n)*(Divide[Pi,Sin[Pi*t]])^(2), {t, - c - I*Infinity, - c + I*Infinity}] Failure Failure Skip
Fail
Complex[1.1891344833338187, 1.6392926414123716] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.6392926414123716, 1.6392926414123716] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.6392926414123716, 1.1891344833338187] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.1891344833338187, 1.1891344833338187] <- {Rule[c, Rational[1, 2]], Rule[BernoulliB[n, x], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Integrate[Times[Power[Pi, 2], Power[Plus[t, x], n], Power[Csc[Times[Pi, t]], 2]], {t, DirectedInfinity[Complex[0, -1]], DirectedInfinity[Complex[0, 1]]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.8.E1 B 2 ⁒ n ⁑ ( x ) = ( - 1 ) n + 1 ⁒ 2 ⁒ ( 2 ⁒ n ) ! ( 2 ⁒ Ο€ ) 2 ⁒ n ⁒ βˆ‘ k = 1 ∞ cos ⁑ ( 2 ⁒ Ο€ ⁒ k ⁒ x ) k 2 ⁒ n Bernoulli-polynomial-B 2 𝑛 π‘₯ superscript 1 𝑛 1 2 2 𝑛 superscript 2 πœ‹ 2 𝑛 superscript subscript π‘˜ 1 2 πœ‹ π‘˜ π‘₯ superscript π‘˜ 2 𝑛 {\displaystyle{\displaystyle B_{2n}\left(x\right)=(-1)^{n+1}\frac{2(2n)!}{(2% \pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos\left(2\pi kx\right)}{k^{2n}}}} bernoulli(2*n, x)=(- 1)^(n + 1)*(2*factorial(2*n))/((2*Pi)^(2*n))*sum((cos(2*Pi*k*x))/((k)^(2*n)), k = 1..infinity) BernoulliB[2*n, x]=(- 1)^(n + 1)*Divide[2*(2*n)!,(2*Pi)^(2*n)]*Sum[Divide[Cos[2*Pi*k*x],(k)^(2*n)], {k, 1, Infinity}] Failure Failure Skip
Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}
6.0 <- {Rule[n, 1], Rule[x, 3]}
4.0 <- {Rule[n, 2], Rule[x, 2]}
36.0 <- {Rule[n, 2], Rule[x, 3]}
... skip entries to safe data
24.8.E2 B 2 ⁒ n + 1 ⁑ ( x ) = ( - 1 ) n + 1 ⁒ 2 ⁒ ( 2 ⁒ n + 1 ) ! ( 2 ⁒ Ο€ ) 2 ⁒ n + 1 ⁒ βˆ‘ k = 1 ∞ sin ⁑ ( 2 ⁒ Ο€ ⁒ k ⁒ x ) k 2 ⁒ n + 1 Bernoulli-polynomial-B 2 𝑛 1 π‘₯ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 πœ‹ 2 𝑛 1 superscript subscript π‘˜ 1 2 πœ‹ π‘˜ π‘₯ superscript π‘˜ 2 𝑛 1 {\displaystyle{\displaystyle B_{2n+1}\left(x\right)=(-1)^{n+1}\frac{2(2n+1)!}{% (2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin\left(2\pi kx\right)}{k^{2n+1}}}} bernoulli(2*n + 1, x)=(- 1)^(n + 1)*(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))*sum((sin(2*Pi*k*x))/((k)^(2*n + 1)), k = 1..infinity) BernoulliB[2*n + 1, x]=(- 1)^(n + 1)*Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]*Sum[Divide[Sin[2*Pi*k*x],(k)^(2*n + 1)], {k, 1, Infinity}] Failure Failure Skip
Fail
3.0 <- {Rule[n, 1], Rule[x, 2]}
15.0 <- {Rule[n, 1], Rule[x, 3]}
5.0 <- {Rule[n, 2], Rule[x, 2]}
85.0 <- {Rule[n, 2], Rule[x, 3]}
... skip entries to safe data
24.8.E4 E 2 ⁒ n ⁑ ( x ) = ( - 1 ) n ⁒ 4 ⁒ ( 2 ⁒ n ) ! Ο€ 2 ⁒ n + 1 ⁒ βˆ‘ k = 0 ∞ sin ⁑ ( ( 2 ⁒ k + 1 ) ⁒ Ο€ ⁒ x ) ( 2 ⁒ k + 1 ) 2 ⁒ n + 1 Euler-polynomial-E 2 𝑛 π‘₯ superscript 1 𝑛 4 2 𝑛 superscript πœ‹ 2 𝑛 1 superscript subscript π‘˜ 0 2 π‘˜ 1 πœ‹ π‘₯ superscript 2 π‘˜ 1 2 𝑛 1 {\displaystyle{\displaystyle E_{2n}\left(x\right)=(-1)^{n}\frac{4(2n)!}{\pi^{2% n+1}}\sum_{k=0}^{\infty}\frac{\sin\left((2k+1)\pi x\right)}{(2k+1)^{2n+1}}}} euler(2*n, x)=(- 1)^(n)*(4*factorial(2*n))/((Pi)^(2*n + 1))*sum((sin((2*k + 1)* Pi*x))/((2*k + 1)^(2*n + 1)), k = 0..infinity) EulerE[2*n, x]=(- 1)^(n)*Divide[4*(2*n)!,(Pi)^(2*n + 1)]*Sum[Divide[Sin[(2*k + 1)* Pi*x],(2*k + 1)^(2*n + 1)], {k, 0, Infinity}] Failure Failure Skip
Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}
6.0 <- {Rule[n, 1], Rule[x, 3]}
2.0 <- {Rule[n, 2], Rule[x, 2]}
30.0 <- {Rule[n, 2], Rule[x, 3]}
... skip entries to safe data
24.8.E5 E 2 ⁒ n - 1 ⁑ ( x ) = ( - 1 ) n ⁒ 4 ⁒ ( 2 ⁒ n - 1 ) ! Ο€ 2 ⁒ n ⁒ βˆ‘ k = 0 ∞ cos ⁑ ( ( 2 ⁒ k + 1 ) ⁒ Ο€ ⁒ x ) ( 2 ⁒ k + 1 ) 2 ⁒ n Euler-polynomial-E 2 𝑛 1 π‘₯ superscript 1 𝑛 4 2 𝑛 1 superscript πœ‹ 2 𝑛 superscript subscript π‘˜ 0 2 π‘˜ 1 πœ‹ π‘₯ superscript 2 π‘˜ 1 2 𝑛 {\displaystyle{\displaystyle E_{2n-1}\left(x\right)=(-1)^{n}\frac{4(2n-1)!}{% \pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos\left((2k+1)\pi x\right)}{(2k+1)^{2n}}}} euler(2*n - 1, x)=(- 1)^(n)*(4*factorial(2*n - 1))/((Pi)^(2*n))*sum((cos((2*k + 1)* Pi*x))/((2*k + 1)^(2*n)), k = 0..infinity) EulerE[2*n - 1, x]=(- 1)^(n)*Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Sum[Divide[Cos[(2*k + 1)* Pi*x],(2*k + 1)^(2*n)], {k, 0, Infinity}] Failure Failure Skip
Fail
2.0 <- {Rule[n, 1], Rule[x, 2]}
2.0 <- {Rule[n, 1], Rule[x, 3]}
2.0 <- {Rule[n, 2], Rule[x, 2]}
14.0 <- {Rule[n, 2], Rule[x, 3]}
... skip entries to safe data
24.8.E6 B 4 ⁒ n + 2 = ( 8 ⁒ n + 4 ) ⁒ βˆ‘ k = 1 ∞ k 4 ⁒ n + 1 e 2 ⁒ Ο€ ⁒ k - 1 Bernoulli-number-B 4 𝑛 2 8 𝑛 4 superscript subscript π‘˜ 1 superscript π‘˜ 4 𝑛 1 superscript 𝑒 2 πœ‹ π‘˜ 1 {\displaystyle{\displaystyle B_{4n+2}=(8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}% {e^{2\pi k}-1}}} bernoulli(4*n + 2)=(8*n + 4)* sum(((k)^(4*n + 1))/(exp(2*Pi*k)- 1), k = 1..infinity) BernoulliB[4*n + 2]=(8*n + 4)* Sum[Divide[(k)^(4*n + 1),Exp[2*Pi*k]- 1], {k, 1, Infinity}] Failure Failure Skip Skip
24.8.E7 B 2 ⁒ n = ( - 1 ) n + 1 ⁒ 4 ⁒ n 2 2 ⁒ n - 1 ⁒ βˆ‘ k = 1 ∞ k 2 ⁒ n - 1 e Ο€ ⁒ k + ( - 1 ) k + n Bernoulli-number-B 2 𝑛 superscript 1 𝑛 1 4 𝑛 superscript 2 2 𝑛 1 superscript subscript π‘˜ 1 superscript π‘˜ 2 𝑛 1 superscript 𝑒 πœ‹ π‘˜ superscript 1 π‘˜ 𝑛 {\displaystyle{\displaystyle B_{2n}=\frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{% \infty}\frac{k^{2n-1}}{e^{\pi k}+(-1)^{k+n}}}} bernoulli(2*n)=((- 1)^(n + 1)* 4*n)/((2)^(2*n)- 1)*sum(((k)^(2*n - 1))/(exp(Pi*k)+(- 1)^(k + n)), k = 1..infinity) BernoulliB[2*n]=Divide[(- 1)^(n + 1)* 4*n,(2)^(2*n)- 1]*Sum[Divide[(k)^(2*n - 1),Exp[Pi*k]+(- 1)^(k + n)], {k, 1, Infinity}] Failure Failure Skip Skip
24.8.E8 B 2 ⁒ n 4 ⁒ n ⁒ ( Ξ± n - ( - Ξ² ) n ) = Ξ± n ⁒ βˆ‘ k = 1 ∞ k 2 ⁒ n - 1 e 2 ⁒ Ξ± ⁒ k - 1 - ( - Ξ² ) n ⁒ βˆ‘ k = 1 ∞ k 2 ⁒ n - 1 e 2 ⁒ Ξ² ⁒ k - 1 Bernoulli-number-B 2 𝑛 4 𝑛 superscript 𝛼 𝑛 superscript 𝛽 𝑛 superscript 𝛼 𝑛 superscript subscript π‘˜ 1 superscript π‘˜ 2 𝑛 1 superscript 𝑒 2 𝛼 π‘˜ 1 superscript 𝛽 𝑛 superscript subscript π‘˜ 1 superscript π‘˜ 2 𝑛 1 superscript 𝑒 2 𝛽 π‘˜ 1 {\displaystyle{\displaystyle\frac{B_{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}% \right)=\alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)% ^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}}} (bernoulli(2*n))/(4*n)*((alpha)^(n)-(- beta)^(n))= (alpha)^(n)* sum(((k)^(2*n - 1))/(exp(2*alpha*k)- 1), k = 1..infinity)-(- beta)^(n)* sum(((k)^(2*n - 1))/(exp(2*beta*k)- 1), k = 1..infinity) Divide[BernoulliB[2*n],4*n]*((\[Alpha])^(n)-(- \[Beta])^(n))= (\[Alpha])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Alpha]*k]- 1], {k, 1, Infinity}]-(- \[Beta])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Beta]*k]- 1], {k, 1, Infinity}] Failure Failure Skip Error
24.8.E9 E 2 ⁒ n = ( - 1 ) n ⁒ βˆ‘ k = 1 ∞ k 2 ⁒ n cosh ⁑ ( 1 2 ⁒ Ο€ ⁒ k ) - 4 ⁒ βˆ‘ k = 0 ∞ ( - 1 ) k ⁒ ( 2 ⁒ k + 1 ) 2 ⁒ n e 2 ⁒ Ο€ ⁒ ( 2 ⁒ k + 1 ) - 1 Euler-number-E 2 𝑛 superscript 1 𝑛 superscript subscript π‘˜ 1 superscript π‘˜ 2 𝑛 1 2 πœ‹ π‘˜ 4 superscript subscript π‘˜ 0 superscript 1 π‘˜ superscript 2 π‘˜ 1 2 𝑛 superscript 𝑒 2 πœ‹ 2 π‘˜ 1 1 {\displaystyle{\displaystyle E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{% \cosh\left(\tfrac{1}{2}\pi k\right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^% {2n}}{e^{2\pi(2k+1)}-1}}} Error EulerE[2*n]=(- 1)^(n)* Sum[Divide[(k)^(2*n),Cosh[Divide[1,2]*Pi*k]], {k, 1, Infinity}]- 4*Sum[Divide[(- 1)^(k)*(2*k + 1)^(2*n),Exp[2*Pi*(2*k + 1)]- 1], {k, 0, Infinity}] Error Failure - Skip
24.9.E1 | B 2 ⁒ n | > | B 2 ⁒ n ⁑ ( x ) | Bernoulli-number-B 2 𝑛 Bernoulli-polynomial-B 2 𝑛 π‘₯ {\displaystyle{\displaystyle|B_{2n}|>|B_{2n}\left(x\right)|}} abs(bernoulli(2*n))>abs(bernoulli(2*n, x)) Abs[BernoulliB[2*n]]>Abs[BernoulliB[2*n, x]] Failure Failure Skip Successful
24.9.E2 ( 2 - 2 1 - 2 ⁒ n ) ⁒ | B 2 ⁒ n | β‰₯ | B 2 ⁒ n ⁑ ( x ) - B 2 ⁒ n | 2 superscript 2 1 2 𝑛 Bernoulli-number-B 2 𝑛 Bernoulli-polynomial-B 2 𝑛 π‘₯ Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle(2-2^{1-2n})|B_{2n}|>=|B_{2n}\left(x\right)-B_{2n}% |}} (2 - (2)^(1 - 2*n))*abs(bernoulli(2*n))> =abs(bernoulli(2*n, x)- bernoulli(2*n)) (2 - (2)^(1 - 2*n))*Abs[BernoulliB[2*n]]> =Abs[BernoulliB[2*n, x]- BernoulliB[2*n]] Failure Failure Skip Successful
24.9.E3 4 - n ⁒ | E 2 ⁒ n | > ( - 1 ) n ⁒ E 2 ⁒ n ⁑ ( x ) superscript 4 𝑛 Euler-number-E 2 𝑛 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 π‘₯ {\displaystyle{\displaystyle 4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)}} Error (4)^(- n)*Abs[EulerE[2*n]]>(- 1)^(n)* EulerE[2*n, x] Error Failure - Successful
24.9.E3 ( - 1 ) n ⁒ E 2 ⁒ n ⁑ ( x ) > 0 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 π‘₯ 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n}\left(x\right)>0}} (- 1)^(n)* euler(2*n, x)> 0 (- 1)^(n)* EulerE[2*n, x]> 0 Failure Failure
Fail
0. < 0. <- {n = 1, x = 1}
0. < -2. <- {n = 1, x = 2}
0. < -6. <- {n = 1, x = 3}
0. < 0. <- {n = 2, x = 1}
... skip entries to safe data
 Successful
24.9.E4 2 ⁒ ( 2 ⁒ n + 1 ) ! ( 2 ⁒ Ο€ ) 2 ⁒ n + 1 > ( - 1 ) n + 1 ⁒ B 2 ⁒ n + 1 ⁑ ( x ) 2 2 𝑛 1 superscript 2 πœ‹ 2 𝑛 1 superscript 1 𝑛 1 Bernoulli-polynomial-B 2 𝑛 1 π‘₯ {\displaystyle{\displaystyle\frac{2(2n+1)!}{(2\pi)^{2n+1}}>(-1)^{n+1}B_{2n+1}% \left(x\right)}} (2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))>(- 1)^(n + 1)* bernoulli(2*n + 1, x) Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]>(- 1)^(n + 1)* BernoulliB[2*n + 1, x] Failure Failure
Fail
3. < .4837730163e-1 <- {n = 1, x = 2}
15. < .4837730163e-1 <- {n = 1, x = 3}
7. < .2607362729e-1 <- {n = 3, x = 2}
455. < .2607362729e-1 <- {n = 3, x = 3}
 Successful
24.9.E4 ( - 1 ) n + 1 ⁒ B 2 ⁒ n + 1 ⁑ ( x ) > 0 superscript 1 𝑛 1 Bernoulli-polynomial-B 2 𝑛 1 π‘₯ 0 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n+1}\left(x\right)>0}} (- 1)^(n + 1)* bernoulli(2*n + 1, x)> 0 (- 1)^(n + 1)* BernoulliB[2*n + 1, x]> 0 Failure Failure
Fail
0. < 0. <- {n = 1, x = 1}
0. < 0. <- {n = 2, x = 1}
0. < -5. <- {n = 2, x = 2}
0. < -85. <- {n = 2, x = 3}
... skip entries to safe data
 Successful
24.9.E5 4 ⁒ ( 2 ⁒ n - 1 ) ! Ο€ 2 ⁒ n ⁒ 2 2 ⁒ n - 1 2 2 ⁒ n - 2 > ( - 1 ) n ⁒ E 2 ⁒ n - 1 ⁑ ( x ) 4 2 𝑛 1 superscript πœ‹ 2 𝑛 superscript 2 2 𝑛 1 superscript 2 2 𝑛 2 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 1 π‘₯ {\displaystyle{\displaystyle\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}% >(-1)^{n}E_{2n-1}\left(x\right)}} (4*factorial(2*n - 1))/((Pi)^(2*n))*((2)^(2*n)- 1)/((2)^(2*n)- 2)>(- 1)^(n)* euler(2*n - 1, x) Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Divide[(2)^(2*n)- 1,(2)^(2*n)- 2]>(- 1)^(n)* EulerE[2*n - 1, x] Failure Failure
Fail
2.250000000 < .2639824007 <- {n = 2, x = 2}
13.75000000 < .2639824007 <- {n = 2, x = 3}
 Successful
24.9.E5 ( - 1 ) n ⁒ E 2 ⁒ n - 1 ⁑ ( x ) > 0 superscript 1 𝑛 Euler-polynomial-E 2 𝑛 1 π‘₯ 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n-1}\left(x\right)>0}} (- 1)^(n)* euler(2*n - 1, x)> 0 (- 1)^(n)* EulerE[2*n - 1, x]> 0 Failure Failure
Fail
0. < -.5000000000 <- {n = 1, x = 1}
0. < -1.500000000 <- {n = 1, x = 2}
0. < -2.500000000 <- {n = 1, x = 3}
0. < -.2500000000 <- {n = 2, x = 1}
... skip entries to safe data
 Successful
24.9.E6 5 ⁒ Ο€ ⁒ n ⁒ ( n Ο€ ⁒ e ) 2 ⁒ n > ( - 1 ) n + 1 ⁒ B 2 ⁒ n 5 πœ‹ 𝑛 superscript 𝑛 πœ‹ 𝑒 2 𝑛 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle 5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-% 1)^{n+1}B_{2n}}} 5*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n)>(- 1)^(n + 1)* bernoulli(2*n) 5*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n)>(- 1)^(n + 1)* BernoulliB[2*n] Failure Failure
Fail
.1666666667 < .1215223702 <- {n = 1}
 Successful
24.9.E6 ( - 1 ) n + 1 ⁒ B 2 ⁒ n > 4 ⁒ Ο€ ⁒ n ⁒ ( n Ο€ ⁒ e ) 2 ⁒ n superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 4 πœ‹ 𝑛 superscript 𝑛 πœ‹ 𝑒 2 𝑛 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>4\sqrt{\pi n}\left(\frac{n}{\pi e% }\right)^{2n}}} (- 1)^(n + 1)* bernoulli(2*n)> 4*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n) (- 1)^(n + 1)* BernoulliB[2*n]> 4*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n) Failure Failure Successful Successful
24.9.E7 8 ⁒ n Ο€ ⁒ ( 4 ⁒ n Ο€ ⁒ e ) 2 ⁒ n ⁒ ( 1 + 1 12 ⁒ n ) > ( - 1 ) n ⁒ E 2 ⁒ n 8 𝑛 πœ‹ superscript 4 𝑛 πœ‹ 𝑒 2 𝑛 1 1 12 𝑛 superscript 1 𝑛 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right% )^{2n}\left(1+\frac{1}{12n}\right)>(-1)^{n}E_{2n}}} Error 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)*(1 +Divide[1,12*n])>(- 1)^(n)* EulerE[2*n] Error Failure - Successful
24.9.E7 ( - 1 ) n ⁒ E 2 ⁒ n > 8 ⁒ n Ο€ ⁒ ( 4 ⁒ n Ο€ ⁒ e ) 2 ⁒ n superscript 1 𝑛 Euler-number-E 2 𝑛 8 𝑛 πœ‹ superscript 4 𝑛 πœ‹ 𝑒 2 𝑛 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n% }{\pi e}\right)^{2n}}} Error (- 1)^(n)* EulerE[2*n]> 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n) Error Failure - Successful
24.9.E8 2 ⁒ ( 2 ⁒ n ) ! ( 2 ⁒ Ο€ ) 2 ⁒ n ⁒ 1 1 - 2 Ξ² - 2 ⁒ n β‰₯ ( - 1 ) n + 1 ⁒ B 2 ⁒ n β‰₯ 2 ⁒ ( 2 ⁒ n ) ! ( 2 ⁒ Ο€ ) 2 ⁒ n ⁒ 1 1 - 2 - 2 ⁒ n 2 2 𝑛 superscript 2 πœ‹ 2 𝑛 1 1 superscript 2 𝛽 2 𝑛 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 2 2 𝑛 superscript 2 πœ‹ 2 𝑛 1 1 superscript 2 2 𝑛 {\displaystyle{\displaystyle\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}}% >=(-1)^{n+1}B_{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}}} (2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(beta - 2*n))> =(- 1)^(n + 1)* bernoulli(2*n)>=(2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(- 2*n)) Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(\[Beta]- 2*n)]> =(- 1)^(n + 1)* BernoulliB[2*n]>=Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(- 2*n)] Failure Failure Error Successful
24.9.E9 Ξ² = 2 + ln ⁑ ( 1 - 6 ⁒ Ο€ - 2 ) ln ⁑ 2 𝛽 2 1 6 superscript πœ‹ 2 2 {\displaystyle{\displaystyle\beta=2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}}} beta = 2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2)) \[Beta]= 2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]] Failure Failure
Fail
.765019736+1.414213562*I <- {beta = 2^(1/2)+I*2^(1/2)}
.765019736-1.414213562*I <- {beta = 2^(1/2)-I*2^(1/2)}
-2.063407388-1.414213562*I <- {beta = -2^(1/2)-I*2^(1/2)}
-2.063407388+1.414213562*I <- {beta = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.7650197375731231, 1.4142135623730951] <- {Rule[Ξ², Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7650197375731231, -1.4142135623730951] <- {Rule[Ξ², Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.0634073871730667, -1.4142135623730951] <- {Rule[Ξ², Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.0634073871730667, 1.4142135623730951] <- {Rule[Ξ², Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
24.9.E10 4 n + 1 ⁒ ( 2 ⁒ n ) ! Ο€ 2 ⁒ n + 1 > ( - 1 ) n ⁒ E 2 ⁒ n superscript 4 𝑛 1 2 𝑛 superscript πœ‹ 2 𝑛 1 superscript 1 𝑛 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}}} Error Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]>(- 1)^(n)* EulerE[2*n] Error Failure - Successful
24.9.E10 ( - 1 ) n ⁒ E 2 ⁒ n > 4 n + 1 ⁒ ( 2 ⁒ n ) ! Ο€ 2 ⁒ n + 1 ⁒ 1 1 + 3 - 1 - 2 ⁒ n superscript 1 𝑛 Euler-number-E 2 𝑛 superscript 4 𝑛 1 2 𝑛 superscript πœ‹ 2 𝑛 1 1 1 superscript 3 1 2 𝑛 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}}} Error (- 1)^(n)* EulerE[2*n]>Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]*Divide[1,1 + (3)^(- 1 - 2*n)] Error Failure - Successful
24.13.E2 ∫ x x + 1 B n ⁑ ( t ) ⁒ d t = x n superscript subscript π‘₯ π‘₯ 1 Bernoulli-polynomial-B 𝑛 𝑑 𝑑 superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\int_{x}^{x+1}B_{n}\left(t\right)\mathrm{d}t=x^{n}}} int(bernoulli(n, t), t = x..x + 1)= (x)^(n) Integrate[BernoulliB[n, t], {t, x, x + 1}]= (x)^(n) Failure Failure Skip Successful
24.13.E3 ∫ x x + ( 1 / 2 ) B n ⁑ ( t ) ⁒ d t = E n ⁑ ( 2 ⁒ x ) 2 n + 1 superscript subscript π‘₯ π‘₯ 1 2 Bernoulli-polynomial-B 𝑛 𝑑 𝑑 Euler-polynomial-E 𝑛 2 π‘₯ superscript 2 𝑛 1 {\displaystyle{\displaystyle\int_{x}^{x+(1/2)}B_{n}\left(t\right)\mathrm{d}t=% \frac{E_{n}\left(2x\right)}{2^{n+1}}}} int(bernoulli(n, t), t = x..x +(1/ 2))=(euler(n, 2*x))/((2)^(n + 1)) Integrate[BernoulliB[n, t], {t, x, x +(1/ 2)}]=Divide[EulerE[n, 2*x],(2)^(n + 1)] Failure Failure Skip Successful
24.13.E4 ∫ 0 1 / 2 B n ⁑ ( t ) ⁒ d t = 1 - 2 n + 1 2 n ⁒ B n + 1 n + 1 superscript subscript 0 1 2 Bernoulli-polynomial-B 𝑛 𝑑 𝑑 1 superscript 2 𝑛 1 superscript 2 𝑛 Bernoulli-number-B 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}B_{n}\left(t\right)\mathrm{d}t=\frac% {1-2^{n+1}}{2^{n}}\frac{B_{n+1}}{n+1}}} int(bernoulli(n, t), t = 0..1/ 2)=(1 - (2)^(n + 1))/((2)^(n))*(bernoulli(n + 1))/(n + 1) Integrate[BernoulliB[n, t], {t, 0, 1/ 2}]=Divide[1 - (2)^(n + 1),(2)^(n)]*Divide[BernoulliB[n + 1],n + 1] Failure Failure Skip Successful
24.13.E5 ∫ 1 / 4 3 / 4 B n ⁑ ( t ) ⁒ d t = E n 2 2 ⁒ n + 1 superscript subscript 1 4 3 4 Bernoulli-polynomial-B 𝑛 𝑑 𝑑 Euler-number-E 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\int_{1/4}^{3/4}B_{n}\left(t\right)\mathrm{d}t=% \frac{E_{n}}{2^{2n+1}}}} Error Integrate[BernoulliB[n, t], {t, 1/ 4, 3/ 4}]=Divide[EulerE[n],(2)^(2*n + 1)] Error Failure - Successful
24.13.E6 ∫ 0 1 B n ⁑ ( t ) ⁒ B m ⁑ ( t ) ⁒ d t = ( - 1 ) n - 1 ⁒ m ! ⁒ n ! ( m + n ) ! ⁒ B m + n superscript subscript 0 1 Bernoulli-polynomial-B 𝑛 𝑑 Bernoulli-polynomial-B π‘š 𝑑 𝑑 superscript 1 𝑛 1 π‘š 𝑛 π‘š 𝑛 Bernoulli-number-B π‘š 𝑛 {\displaystyle{\displaystyle\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)% \mathrm{d}t=\frac{(-1)^{n-1}m!n!}{(m+n)!}B_{m+n}}} int(bernoulli(n, t)*bernoulli(m, t), t = 0..1)=((- 1)^(n - 1)* factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n) Integrate[BernoulliB[n, t]*BernoulliB[m, t], {t, 0, 1}]=Divide[(- 1)^(n - 1)* (m)!*(n)!,(m + n)!]*BernoulliB[m + n] Failure Failure Skip Successful
24.13.E8 ∫ 0 1 E n ⁑ ( t ) ⁒ d t = - 2 ⁒ E n + 1 ⁑ ( 0 ) n + 1 superscript subscript 0 1 Euler-polynomial-E 𝑛 𝑑 𝑑 2 Euler-polynomial-E 𝑛 1 0 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1}E_{n}\left(t\right)\mathrm{d}t=-2\frac% {E_{n+1}\left(0\right)}{n+1}}} int(euler(n, t), t = 0..1)= - 2*(euler(n + 1, 0))/(n + 1) Integrate[EulerE[n, t], {t, 0, 1}]= - 2*Divide[EulerE[n + 1, 0],n + 1] Failure Failure Skip Successful
24.13.E8 - 2 ⁒ E n + 1 ⁑ ( 0 ) n + 1 = 4 ⁒ ( 2 n + 2 - 1 ) ( n + 1 ) ⁒ ( n + 2 ) ⁒ B n + 2 2 Euler-polynomial-E 𝑛 1 0 𝑛 1 4 superscript 2 𝑛 2 1 𝑛 1 𝑛 2 Bernoulli-number-B 𝑛 2 {\displaystyle{\displaystyle-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2% }-1)}{(n+1)(n+2)}B_{n+2}}} - 2*(euler(n + 1, 0))/(n + 1)=(4*((2)^(n + 2)- 1))/((n + 1)*(n + 2))*bernoulli(n + 2) - 2*Divide[EulerE[n + 1, 0],n + 1]=Divide[4*((2)^(n + 2)- 1),(n + 1)*(n + 2)]*BernoulliB[n + 2] Failure Failure Successful Successful
24.13.E9 ∫ 0 1 / 2 E 2 ⁒ n ⁑ ( t ) ⁒ d t = - E 2 ⁒ n + 1 ⁑ ( 0 ) 2 ⁒ n + 1 superscript subscript 0 1 2 Euler-polynomial-E 2 𝑛 𝑑 𝑑 Euler-polynomial-E 2 𝑛 1 0 2 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}E_{2n}\left(t\right)\mathrm{d}t=-% \frac{E_{2n+1}\left(0\right)}{2n+1}}} int(euler(2*n, t), t = 0..1/ 2)= -(euler(2*n + 1, 0))/(2*n + 1) Integrate[EulerE[2*n, t], {t, 0, 1/ 2}]= -Divide[EulerE[2*n + 1, 0],2*n + 1] Failure Failure Skip Successful
24.13.E9 - E 2 ⁒ n + 1 ⁑ ( 0 ) 2 ⁒ n + 1 = 2 ⁒ ( 2 2 ⁒ n + 2 - 1 ) ⁒ B 2 ⁒ n + 2 ( 2 ⁒ n + 1 ) ⁒ ( 2 ⁒ n + 2 ) Euler-polynomial-E 2 𝑛 1 0 2 𝑛 1 2 superscript 2 2 𝑛 2 1 Bernoulli-number-B 2 𝑛 2 2 𝑛 1 2 𝑛 2 {\displaystyle{\displaystyle-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n% +2}-1)B_{2n+2}}{(2n+1)(2n+2)}}} -(euler(2*n + 1, 0))/(2*n + 1)=(2*((2)^(2*n + 2)- 1)* bernoulli(2*n + 2))/((2*n + 1)*(2*n + 2)) -Divide[EulerE[2*n + 1, 0],2*n + 1]=Divide[2*((2)^(2*n + 2)- 1)* BernoulliB[2*n + 2],(2*n + 1)*(2*n + 2)] Failure Failure Successful Successful
24.13.E10 ∫ 0 1 / 2 E 2 ⁒ n - 1 ⁑ ( t ) ⁒ d t = E 2 ⁒ n n ⁒ 2 2 ⁒ n + 1 superscript subscript 0 1 2 Euler-polynomial-E 2 𝑛 1 𝑑 𝑑 Euler-number-E 2 𝑛 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\int_{0}^{1/2}E_{2n-1}\left(t\right)\mathrm{d}t=% \frac{E_{2n}}{n2^{2n+1}}}} Error Integrate[EulerE[2*n - 1, t], {t, 0, 1/ 2}]=Divide[EulerE[2*n],n*(2)^(2*n + 1)] Error Failure - Successful
24.13.E11 ∫ 0 1 E n ⁑ ( t ) ⁒ E m ⁑ ( t ) ⁒ d t = ( - 1 ) n ⁒ 4 ⁒ ( 2 m + n + 2 - 1 ) ⁒ m ! ⁒ n ! ( m + n + 2 ) ! ⁒ B m + n + 2 superscript subscript 0 1 Euler-polynomial-E 𝑛 𝑑 Euler-polynomial-E π‘š 𝑑 𝑑 superscript 1 𝑛 4 superscript 2 π‘š 𝑛 2 1 π‘š 𝑛 π‘š 𝑛 2 Bernoulli-number-B π‘š 𝑛 2 {\displaystyle{\displaystyle\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)% \mathrm{d}t=(-1)^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}}} int(euler(n, t)*euler(m, t), t = 0..1)=(- 1)^(n)* 4*(((2)^(m + n + 2)- 1)* factorial(m)*factorial(n))/(factorial(m + n + 2))*bernoulli(m + n + 2) Integrate[EulerE[n, t]*EulerE[m, t], {t, 0, 1}]=(- 1)^(n)* 4*Divide[((2)^(m + n + 2)- 1)* (m)!*(n)!,(m + n + 2)!]*BernoulliB[m + n + 2] Failure Failure Skip Successful
24.14.E4 - 2 n + 1 ⁒ E n + 1 ⁑ ( 0 ) = - 2 n + 2 ⁒ ( 1 - 2 n + 2 ) ⁒ B n + 2 n + 2 superscript 2 𝑛 1 Euler-polynomial-E 𝑛 1 0 superscript 2 𝑛 2 1 superscript 2 𝑛 2 Bernoulli-number-B 𝑛 2 𝑛 2 {\displaystyle{\displaystyle-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(1-2^{n+2})% \frac{B_{n+2}}{n+2}}} - (2)^(n + 1)* euler(n + 1, 0)= - (2)^(n + 2)*(1 - (2)^(n + 2))*(bernoulli(n + 2))/(n + 2) - (2)^(n + 1)* EulerE[n + 1, 0]= - (2)^(n + 2)*(1 - (2)^(n + 2))*Divide[BernoulliB[n + 2],n + 2] Failure Failure Successful Successful
24.14.E7 βˆ‘ j = 0 m βˆ‘ k = 0 n ( m j ) ⁒ ( n k ) ⁒ B j ⁒ B k m + n - j - k + 1 = ( - 1 ) m - 1 ⁒ m ! ⁒ n ! ( m + n ) ! ⁒ B m + n superscript subscript 𝑗 0 π‘š superscript subscript π‘˜ 0 𝑛 binomial π‘š 𝑗 binomial 𝑛 π‘˜ Bernoulli-number-B 𝑗 Bernoulli-number-B π‘˜ π‘š 𝑛 𝑗 π‘˜ 1 superscript 1 π‘š 1 π‘š 𝑛 π‘š 𝑛 Bernoulli-number-B π‘š 𝑛 {\displaystyle{\displaystyle\sum_{j=0}^{m}\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{% }{m}{j}\genfrac{(}{)}{0.0pt}{}{n}{k}\frac{B_{j}B_{k}}{m+n-j-k+1}=(-1)^{m-1}% \frac{m!n!}{(m+n)!}B_{m+n}}} sum(sum(binomial(m,j)*binomial(n,k)*(bernoulli(j)*bernoulli(k))/(m + n - j - k + 1), k = 0..n), j = 0..m)=(- 1)^(m - 1)*(factorial(m)*factorial(n))/(factorial(m + n))*bernoulli(m + n) Sum[Sum[Binomial[m,j]*Binomial[n,k]*Divide[BernoulliB[j]*BernoulliB[k],m + n - j - k + 1], {k, 0, n}], {j, 0, m}]=(- 1)^(m - 1)*Divide[(m)!*(n)!,(m + n)!]*BernoulliB[m + n] Failure Failure Skip Successful
24.15.E2 G n = 2 ⁒ ( 1 - 2 n ) ⁒ B n subscript 𝐺 𝑛 2 1 superscript 2 𝑛 Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle G_{n}=2(1-2^{n})B_{n}}} G[n]= 2*(1 - (2)^(n))* bernoulli(n) Subscript[G, n]= 2*(1 - (2)^(n))* BernoulliB[n] Failure Failure
Fail
.414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 1}
2.414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 2}
1.414213562+1.414213562*I <- {G[n] = 2^(1/2)+I*2^(1/2), n = 3}
.414213562-1.414213562*I <- {G[n] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Successful
24.15.E3 tan ⁑ t = βˆ‘ n = 0 ∞ T n ⁒ t n n ! 𝑑 superscript subscript 𝑛 0 subscript 𝑇 𝑛 superscript 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle\tan t=\sum_{n=0}^{\infty}T_{n}\frac{t^{n}}{n!}}} tan(t)= sum(T[n]*((t)^(n))/(factorial(n)), n = 0..infinity) Tan[t]= Sum[Subscript[T, n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}] Failure Failure Skip Skip
24.15.E4 T 2 ⁒ n - 1 = ( - 1 ) n - 1 ⁒ 2 2 ⁒ n ⁒ ( 2 2 ⁒ n - 1 ) 2 ⁒ n ⁒ B 2 ⁒ n subscript 𝑇 2 𝑛 1 superscript 1 𝑛 1 superscript 2 2 𝑛 superscript 2 2 𝑛 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{% 2n}}} T[2*n - 1]=(- 1)^(n - 1)*((2)^(2*n)*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n) Subscript[T, 2*n - 1]=(- 1)^(n - 1)*Divide[(2)^(2*n)*((2)^(2*n)- 1),2*n]*BernoulliB[2*n] Failure Failure
Fail
.414213562+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 1}
-.585786438+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 2}
-14.58578644+1.414213562*I <- {T[2*n-1] = 2^(1/2)+I*2^(1/2), n = 3}
.414213562-1.414213562*I <- {T[2*n-1] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
 Successful
24.15.E6 B n = βˆ‘ k = 0 n ( - 1 ) k ⁒ k ! ⁒ S ⁑ ( n , k ) k + 1 Bernoulli-number-B 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ π‘˜ Stirling-number-second-kind-S 𝑛 π‘˜ π‘˜ 1 {\displaystyle{\displaystyle B_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{k!S\left(n,k% \right)}{k+1}}} bernoulli(n)= sum((- 1)^(k)*(factorial(k)*Stirling2(n, k))/(k + 1), k = 0..n) BernoulliB[n]= Sum[(- 1)^(k)*Divide[(k)!*StirlingS2[n, k],k + 1], {k, 0, n}] Failure Successful Skip -
24.15.E7 B n = βˆ‘ k = 0 n ( - 1 ) k ⁒ ( n + 1 k + 1 ) ⁒ S ⁑ ( n + k , k ) / ( n + k k ) Bernoulli-number-B 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ binomial 𝑛 1 π‘˜ 1 Stirling-number-second-kind-S 𝑛 π‘˜ π‘˜ binomial 𝑛 π‘˜ π‘˜ {\displaystyle{\displaystyle B_{n}=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}% {}{n+1}{k+1}S\left(n+k,k\right)\bigg{/}\genfrac{(}{)}{0.0pt}{}{n+k}{k}}} bernoulli(n)= sum((- 1)^(k)*binomial(n + 1,k + 1)*Stirling2(n + k, k)/binomial(n + k,k), k = 0..n) BernoulliB[n]= Sum[(- 1)^(k)*Binomial[n + 1,k + 1]*StirlingS2[n + k, k]/Binomial[n + k,k], {k, 0, n}] Failure Failure Skip
Fail
Complex[0.0, 2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, 2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
2.8284271247461903 <- {Rule[BernoulliB[n], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, -2.8284271247461903] <- {Rule[BernoulliB[n], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Binomial[Plus[1, n], Plus[1, k]], Power[Binomial[Plus[k, n], k], -1], StirlingS2[Plus[k, n], k]], {k, 0, n}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.15.E8 βˆ‘ k = 0 n ( - 1 ) n + k ⁒ s ⁑ ( n + 1 , k + 1 ) ⁒ B k = n ! n + 1 superscript subscript π‘˜ 0 𝑛 superscript 1 𝑛 π‘˜ Stirling-number-first-kind-S 𝑛 1 π‘˜ 1 Bernoulli-number-B π‘˜ 𝑛 𝑛 1 {\displaystyle{\displaystyle\sum_{k=0}^{n}(-1)^{n+k}s\left(n+1,k+1\right)B_{k}% =\frac{n!}{n+1}}} sum((- 1)^(n + k)* Stirling1(n + 1, k + 1)*bernoulli(k), k = 0..n)=(factorial(n))/(n + 1) Sum[(- 1)^(n + k)* StirlingS1[n + 1, k + 1]*BernoulliB[k], {k, 0, n}]=Divide[(n)!,n + 1] Failure Failure Skip Successful
24.16.E5 t ln ⁑ ( 1 + t ) = βˆ‘ n = 0 ∞ b n ⁒ t n 𝑑 1 𝑑 superscript subscript 𝑛 0 subscript 𝑏 𝑛 superscript 𝑑 𝑛 {\displaystyle{\displaystyle\frac{t}{\ln\left(1+t\right)}=\sum_{n=0}^{\infty}b% _{n}t^{n}}} (t)/(ln(1 + t))= sum(b[n]*(t)^(n), n = 0..infinity) Divide[t,Log[1 + t]]= Sum[Subscript[b, n]*(t)^(n), {n, 0, Infinity}] Failure Failure Skip Skip
24.16.E8 Ξ² n ⁒ ( Ξ» ) = n ! ⁒ b n ⁒ Ξ» n + βˆ‘ k = 1 ⌊ n / 2 βŒ‹ n 2 ⁒ k ⁒ B 2 ⁒ k ⁒ s ⁑ ( n - 1 , 2 ⁒ k - 1 ) ⁒ Ξ» n - 2 ⁒ k subscript 𝛽 𝑛 πœ† 𝑛 subscript 𝑏 𝑛 superscript πœ† 𝑛 superscript subscript π‘˜ 1 𝑛 2 𝑛 2 π‘˜ Bernoulli-number-B 2 π‘˜ Stirling-number-first-kind-S 𝑛 1 2 π‘˜ 1 superscript πœ† 𝑛 2 π‘˜ {\displaystyle{\displaystyle\beta_{n}(\lambda)=n!b_{n}\lambda^{n}+\sum_{k=1}^{% \left\lfloor\ifrac{n}{2}\right\rfloor}\frac{n}{2k}B_{2k}s\left(n-1,2k-1\right)% \lambda^{n-2k}}} beta[n]*(lambda)= factorial(n)*b[n]*(lambda)^(n)+ sum((n)/(2*k)*bernoulli(2*k)*Stirling1(n - 1, 2*k - 1)*(lambda)^(n - 2*k), k = 1..floor((n)/(2))) Subscript[\[Beta], n]*(\[Lambda])= (n)!*Subscript[b, n]*(\[Lambda])^(n)+ Sum[Divide[n,2*k]*BernoulliB[2*k]*StirlingS1[n - 1, 2*k - 1]*(\[Lambda])^(n - 2*k), {k, 1, Floor[Divide[n,2]]}] Failure Failure Skip Skip
24.16.E12 B n ⁑ ( x ) = B n , Ο‡ 0 ⁒ ( x - 1 ) Bernoulli-polynomial-B 𝑛 π‘₯ subscript 𝐡 𝑛 subscript πœ’ 0 π‘₯ 1 {\displaystyle{\displaystyle B_{n}\left(x\right)=B_{n,\chi_{0}}(x-1)}} bernoulli(n, x)= B[n , chi[0]]*(x - 1) BernoulliB[n, x]= Subscript[B, n , Subscript[\[Chi], 0]]*(x - 1) Failure Failure
Fail
.5000000000 <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
.85786438e-1-1.414213562*I <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
-.328427124-2.828427124*I <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
.1666666667 <- {B[n,chi[0]] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
Fail
0.5 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1], Rule[Subscript[Ο‡, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.16666666666666666 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1], Rule[Subscript[Ο‡, 0], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
0.5 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 1], Rule[x, 1], Rule[Subscript[Ο‡, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
0.16666666666666666 <- {Rule[B, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[n, 2], Rule[x, 1], Rule[Subscript[Ο‡, 0], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.16.E13 E n ⁑ ( x ) = - 2 1 - n n + 1 ⁒ B n + 1 , Ο‡ 4 ⁒ ( 2 ⁒ x - 1 ) Euler-polynomial-E 𝑛 π‘₯ superscript 2 1 𝑛 𝑛 1 subscript 𝐡 𝑛 1 subscript πœ’ 4 2 π‘₯ 1 {\displaystyle{\displaystyle E_{n}\left(x\right)=-\frac{2^{1-n}}{n+1}B_{n+1,% \chi_{4}}(2x-1)}} euler(n, x)= -((2)^(1 - n))/(n + 1)*B[n + 1 , chi[4]]*(2*x - 1) EulerE[n, x]= -Divide[(2)^(1 - n),n + 1]*Subscript[B, n + 1 , Subscript[\[Chi], 4]]*(2*x - 1) Failure Failure
Fail
1.207106781+.7071067810*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 1}
3.621320343+2.121320343*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 2}
6.035533905+3.535533905*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 1, x = 3}
.2357022604+.2357022604*I <- {B[n+1,chi[4]] = 2^(1/2)+I*2^(1/2), n = 2, x = 1}
... skip entries to safe data
 Successful
24.19#Ex2 B 2 ⁒ n = N 2 ⁒ n D 2 ⁒ n Bernoulli-number-B 2 𝑛 subscript 𝑁 2 𝑛 subscript 𝐷 2 𝑛 {\displaystyle{\displaystyle B_{2n}=\dfrac{N_{2n}}{D_{2n}}}} bernoulli(2*n)=(N[2*n])/(D[2*n]) BernoulliB[2*n]=Divide[Subscript[N, 2*n],Subscript[D, 2*n]] Failure Failure
Fail
-.8333333333 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 1}
-1.033333333 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 2}
-.9761904762 <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)+I*2^(1/2), n = 3}
.1666666667+1.000000000*I <- {D[2*n] = 2^(1/2)+I*2^(1/2), N[2*n] = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, 2.414213562373095] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.414213562373095, 1.4142135623730951] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, 0.41421356237309515] <- {Rule[BernoulliB[Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[D, Times[2, n]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[N, Times[2, n]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
24.19.E3 t 2 cosh ⁑ t - 1 = - 2 ⁒ βˆ‘ n = 0 ∞ ( 2 ⁒ n - 1 ) ⁒ B 2 ⁒ n ⁒ t 2 ⁒ n ( 2 ⁒ n ) ! superscript 𝑑 2 𝑑 1 2 superscript subscript 𝑛 0 2 𝑛 1 Bernoulli-number-B 2 𝑛 superscript 𝑑 2 𝑛 2 𝑛 {\displaystyle{\displaystyle\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1% )B_{2n}\frac{t^{2n}}{(2n)!}}} ((t)^(2))/(cosh(t)- 1)= - 2*sum((2*n - 1)* bernoulli(2*n)*((t)^(2*n))/(factorial(2*n)), n = 0..infinity) Divide[(t)^(2),Cosh[t]- 1]= - 2*Sum[(2*n - 1)* BernoulliB[2*n]*Divide[(t)^(2*n),(2*n)!], {n, 0, Infinity}] Failure Failure Skip Error
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