Alternate Ending

Calculus Level 5

\[\frac{1}{1^3}-\frac{2}{1^3+2^3}+\frac{3}{1^3+2^3+3^3}-\frac{4}{1^3+2^3+3^3+4^3}+\cdots\] If the given sum equal to \[\dfrac{{\pi}^A}{B}+C\ln{A}-C\] for positive integers \(A\), \(B\), and \(C\), find \(A+B+C\).