How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

Question

How can one prove this identity?

$$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$

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There is a formula for $\zeta$ values at even integers, but it involves Bernoulli numbers; simply plugging it in does not appear to be an efficient approach.

Answer 1

Since

$$\zeta(2n) = \frac{1}{(2n-1)!}\int_{0}^{\infty}\frac{x^{2n-1}}{e^x-1}\,dx $$

we have:

$$\sum_{n=1}^{\infty}\frac{\zeta(2n)}{2^{2n-1}} = \int_{0}^{\infty}\frac{\sinh(x/2)}{e^x-1}\,dx =\frac12\int_{0}^{\infty}e^{-x/2}\,dx = \color{red}{1}.$$

Answer 2

$$ \begin{align} \sum_{n=1}^\infty\frac{\zeta(2n)}{2^{2n-1}} &=\sum_{n=1}^\infty\sum_{k=1}^\infty\frac2{k^{2n}2^{2n}}\tag{1}\\ &=2\sum_{k=1}^\infty\sum_{n=1}^\infty\frac1{(4k^2)^n}\tag{2}\\ &=2\sum_{k=1}^\infty\frac1{4k^2-1}\tag{3}\\ &=\sum_{k=1}^\infty\left(\frac1{2k-1}-\frac1{2k+1}\right)\tag{4}\\[6pt] &=1\tag{5} \end{align} $$ Explanation:
$\text{(1):}$ expand $\zeta(2n)=\sum\limits_{k=1}^\infty\frac1{k^{2n}}$
$\text{(2):}$ change the order of summation
$\phantom{\text{(2):}}$ allowed because the terms are all positive
$\text{(3):}$ sum of a geometric series
$\text{(4):}$ partial fractions
$\text{(5):}$ telescoping sum

Answer 3

The Laurent expansion of $\cot (z)$ at the origin in terms of the Riemann zeta function is $$ \cot (z) = \frac{1}{z} - 2 \sum_{k=1}^{\infty}\zeta(2k) \frac{z^{2k-1}}{\pi^{2k}} \ , \ 0 < |z| < \pi. $$

Letting $ \displaystyle z= \frac{\pi}{2}$, $$\cot \left(\frac{\pi}{2} \right) = \frac{2}{\pi} - \frac{2}{\pi} \sum_{k=1}^{\infty} \frac{\zeta(2k)}{2^{2k-1}}.$$

But $\cot \left(\frac{\pi}{2} \right)=0$.

Therefore,

$$ \sum_{k=1}^{\infty} \frac{\zeta(2k)}{2^{2k-1}} = 1.$$

Answer 4

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\begin{align} &\bbox[10px,#ffd]{\sum_{n = 1}^{\infty}{\zeta\pars{2n} \over 2^{2n - 1}}} = \sum_{n = 2}^{\infty}{\zeta\pars{n} \over 2^{n - 1}} - \sum_{n = 1}^{\infty}{\zeta\pars{2n + 1} \over 2^{2n}} \\[3mm] = &\ -\sum_{n = 2}^{\infty}\pars{-1}^{n}\zeta\pars{n}\pars{-\,\half}^{n - 1} - \sum_{n = 1}^{\infty}\bracks{\zeta\pars{2n + 1} - 1}\pars{\half}^{2n}\ -\ \underbrace{\sum_{n = 1}^{\infty}\pars{\half}^{2n}}_{\ds{1 \over 3}} \\ = &\ -\bracks{\Psi\pars{1 + z} + \gamma}_{\ z\ =\ -1/2} \\[3mm] & - \bracks{% {1 \over 2z} - \half\,\pi\cot\pars{\pi z} - {1 \over 1 - z^{2}} + 1 - \gamma - \Psi\pars{1 + z}}_{\ z\ =\ 1/2} - {1 \over 3} \\[8mm] = &\ -\Psi\pars{\half} - {2 \over 3}\ +\ \underbrace{\Psi\pars{3 \over 2}}_{\ds{\Psi\pars{1/2} + 1/\pars{1/2}}} - {1 \over 3} = \color{#f00}{1} \end{align}

$\Psi$ and $\gamma$ are the Digamma function and the Euler-Mascheroni constant, respectively. We used the Digamma Recurrence Formula $\ds{\Psi\pars{z + 1} = \Psi\pars{z} + 1/z}$ and the identities $\mathbf{6.3.14}$ and $\mathbf{6.3.15}$ in this link.