Logs as an infinite sum

Calculus Level 4

∑ n = 2 ∞ ( − 1 ) n n − n 2 \large \sum_{n=2}^\infty \dfrac{(-1)^n}{n-n^2}

If the series above is in the form of A − B ln ⁡ C A - B\ln C , where A , B , C A,B,C are positive integers with C C square-free, find A + B + C A+B+C .


The answer is 5.

Hamza A by Hamza A , 19, USA

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1 solution

Use taylor series for ln(1+x). Then divide both sides by x^2 and then integrate with limit from 0 to 1. You will not run into any problem as after integration lnx will cancel out from both sides . and we know lim (ln(1+x))/x for x tends to 0 is 1. So the result after doing that will be.
1-2ln2. Sorry i could not post the entire solution as i am a noob and i cannot write codes for mathematical symbols. Also my internet is very slow to open websites that lets you copy paste mathematical symbols.

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