FoxTrot

Calculus Level 5

$\large \sum\limits_{k=1}^{\infty} \frac{(-1)^{k+1}k^2}{k^3+1}=\frac{1}{A}\left[1-\ln(B)+\pi \, \text{sech}\left(\frac{\sqrt{3}\:\pi}{C}\right)\right]$

Find the value of $$A+B+C$$, where $$A,B$$ and $$C$$ are positive integers.

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