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Find-lim-n-H-n-k-1-n-1-n-k-k-1-n-cot-1-n-k-




Question Number 190021 by Shrinava last updated on 26/Mar/23
Find:  lim_(n→∞)  H_n  (Σ_(k=1) ^n  (1/(n+k)) -Σ_(k=1) ^n  cot^(-1)  (n+k))
$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{H}_{\boldsymbol{\mathrm{n}}} \:\left(\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}\:-\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{cot}^{-\mathrm{1}} \:\left(\mathrm{n}+\mathrm{k}\right)\right) \\ $$

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