Inverse Sum Cubed

Calculus Level 5

$\large\displaystyle\sum_{n=1}^{\infty} \left({\displaystyle\sum_{k=0}^nk^3} \right)^{-1}$

If the value of the series above can be expressed as $\dfrac{a\pi^2}b-c$ where $a,b,c$ are positive integers, and $a,b$ are coprime to each other, find the value of $a+b+c$ .

See Also Inverse Sum and Inverse Sum Squared.