Formula:DLMF:25.11:E39

<< Formula:DLMF:25.11:E38

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E41 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}\HurwitzZeta@{k+1}{\tfrac{3}{4}}=8C}}}$

Note(s)

${\displaystyle{\displaystyle{\displaystyle G}}}$ is \emph{Catalan's constant} & ${\displaystyle{\displaystyle{\displaystyle{\displaystyle C=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle G}}}$ : Catalan's constant : http://dlmf.nist.gov/25.11.E40
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7

Bibliography

Equation (39), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E41 >>