Let’s rewrite it first, we want to determine
[math]\displaystyle S=\sum_{n=1}^{+\infty}\frac{(-1)^{n+1}}{(n+1)!-n!}=\sum_{n=1}^{+\infty}\frac{(-1)^{n+1}}{nn!}[/math]
Note that I misread the question. This answer tackles a similar problem, but the original sum is solved in Siddhartha Ganguly's ...
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