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