Infinite Series $\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$

Question

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.

Answer 1

Using Maple I am obtaining

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