I know alternating series test so I can figure out this series converge. But I don't know the result of this infinity series. I try to solve this problem in various way, for example taylor series, abel's partial summation formula, and harmonic number but still I can't solve it. I need your help. Thanks very much.
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Can you write down the sum in its expanded form then it would be helpful? – Rounak Sarkar Jul 28 '21 at 13:22
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Note that the multiples summands are simply $H_{n+1}$, where $H_{n}$ denotes the $n$-th harmonic number. – vitamin d Jul 28 '21 at 13:24
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You could begin by simplifying: $\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}(1+1/2+…+1/n+1/(n+1))}{n+2} = \sum\limits_{n=2}^{\infty}\frac{(-1)^n(1+1/2+…+1/n)}{n+1}$ – jjagmath Jul 28 '21 at 13:30