Sum them all up for the second time!

Calculus Level 5

$\large \sum _{ x=1 }^{ \infty }{ \dfrac { \psi ( x ) }{ { x }^{ 2 } } } =\zeta ( A ) -\dfrac { \gamma { \pi }^{ B } }{ C }$

If the equation above holds true for integers $A,B$ and $C$ , find $A+B+C$ .

Notation : $\psi(\cdot)$ denotes the Digamma function .

1 solution

Aditya Narayan Sharma
May 31, 2016

$\displaystyle \psi(n) = H_{n-1} - \gamma$

$\displaystyle \sum_{n=1}^{\infty} \frac{\psi(n)}{n^2} = \sum_{n=1}^{\infty} \frac{H_{n-1}}{n^2} - \gamma\zeta(2)$

$\displaystyle \sum_{n=1}^{\infty} \frac{\psi(n)}{n^2} = s_{h}(1,2) - \frac{\gamma\pi^2}{6} = \zeta(3)-\frac{\gamma\pi^2}{6}$

Thus the answer : $\boxed{3+6+2=11}$