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

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