Alternate Ending

Calculus Level 5

113213+23+313+23+33413+23+33+43+\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 πAB+ClnAC\dfrac{{\pi}^A}{B}+C\ln{A}-C for positive integers AA, BB, and CC, find A+B+CA+B+C.

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