Calculus Level 5

$S=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\dfrac{(-1)^{i+j+k+l}}{i+j+k+l}=\log(a)-\dfrac{a}{b}$ $$a,b$$ are co-prime positive integers. Find $$a+b$$.

NOTE : $$S$$ does converge.

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