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 $A+B+C$ .