If-H-n-1-1-2-1-3-1-n-1-n-prove-that-n-1-H-n-ln-2-n-ln-2-2-

Question Number 211594 by mnjuly1970 last updated on 14/Sep/24

I f, {\bar{H}}_{n} = 1 - \frac{1}{2} + \frac{1}{3} - \dots + \frac{{(- 1)}^{n + 1}}{n}

p r o v e t h a t : \underset{n = 1}{\sum^{\infty}} \frac{{\bar{H}}_{n} - \ln (2)}{n} = \ln^{2} (2)

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Commented by Ghisom last updated on 14/Sep/24

just a hint :

{\bar{H}}_{n} = \underset{j = 1}{\sum^{n}} \frac{{(- 1)}^{j + 1}}{j} \Rightarrow \lim_{n \to \infty} {\bar{H}}_{n} = \ln 2

[Taylor series of \ln (x + 1); ∣ x ∣< 1 \lor x = 1]

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