Tinkering With The $\zeta$ Function

Calculus Level 5

$\large \sum_{n=1}^{\infty} \left(\dfrac{\zeta(2n)}{n} - \dfrac{1}{2n} - \dfrac{1}{2n+1} \right)$

Find the value of the closed form of the above series.

Give your answer to 1 decimal place.

Notation: $\zeta(\cdot)$ denotes the Riemann zeta function.

Tinkering With The ζ\zetaζ Function