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lim-n-k-1-n-arctan-1-2k-2-arctan-2k-2-1-2k-2-




Question Number 151956 by mathdanisur last updated on 24/Aug/21
lim_(n→∞)  Σ_(k=1) ^n  arctan (1/(2k^2 )) arctan ((2k^2  - 1)/(2k^2 )) = ?
$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\mathrm{arctan}\:\frac{\mathrm{1}}{\mathrm{2k}^{\mathrm{2}} }\:\mathrm{arctan}\:\frac{\mathrm{2k}^{\mathrm{2}} \:-\:\mathrm{1}}{\mathrm{2k}^{\mathrm{2}} }\:=\:? \\ $$

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