Math Help - Summation of an Infinite Series

1. Summation of an Infinite Series

Show that

$\sum_{n=1}^\infty \frac{\zeta(2n)}{n 2^{2n}} = \ln \left( \frac{\pi}{2}\right)$

2. Re: Summation of an Infinite Series

I used the equality : $\Gamma(s) \zeta(s) = \int^{\infty}_0 \frac{t^{s-1}}{e^t-1}\,dt$

and got $\,2\int^{\infty}_0 \frac{\cosh(\frac{t}{2})-1}{t(e^t-1)}\, dt$

I am still trying to find this integral ,