Infinite Series $\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$ Evaluate:
$$\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$$
Recognizing that $\zeta(2)-\sum_{n=1}^k\frac1{n^2}$ can be written as $\psi_1(1+k)$ where $\psi_1(z)$ is the trigamma function,
What remains to be done is to evaluate:
$$\sum_{k=1}^\infty\psi_1^2(k+1)$$
Mathematica could not evaluate it in a closed form but the source assures that it exists.
If you liked this problem check out Hard Definite integral involving the Zeta function.

Using Maple I am obtaining
$$\sum _{k=1}^{\infty } \left( \Psi \left( 1,k+1 \right)  \right) ^{2}=
 0.9003626252$$
