Question Number 57848 by Abdo msup. last updated on 13/Apr/19
$$\left.\mathrm{1}\right){prove}\:{that}\:{arctan}\left({a}\right)\:+{arctanb}\:={arctan}\left(\frac{{a}+{b}}{\mathrm{1}−{ab}}\right)\: \\ $$$${with}\:{ab}\neq\mathrm{1} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:{S}_{{N}} =\:\sum_{{n}=\mathrm{1}} ^{{N}} \left(−\mathrm{1}\right)^{{n}} \:{arctan}\left(\frac{\mathrm{2}{n}+\mathrm{1}}{{n}^{\mathrm{2}} \:+{n}−\mathrm{1}}\right) \\ $$
Commented by maxmathsup by imad last updated on 13/Apr/19
$$\left.\mathrm{1}\right)\:{let}\:{put}\:\alpha\:={arctan}\left({a}\right)\:{and}\:\beta\:={arctan}\left({b}\right)\:\:\Rightarrow \\ $$$${tan}\left(\alpha+\beta\right)\:=\frac{{tan}\left(\alpha\right)+{tan}\left(\beta\right)}{\mathrm{1}−{tan}\left(\alpha\right){tan}\left(\beta\right)}\:=\frac{{a}+{b}}{\mathrm{1}−{ab}}\:\:\:\left({we}\:{suppose}\:{ab}\:\neq\mathrm{1}\right)\:\Rightarrow \\ $$$$\alpha+\beta\:={arctan}\left(\frac{{a}+{b}}{\mathrm{1}−{ab}}\right) \\ $$$$\left.\mathrm{2}\right)\:{we}\:{have}\:{S}_{{n}} =−\sum_{{n}=\mathrm{1}} ^{{N}} \:\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left(\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{1}−{n}\left({n}+\mathrm{1}\right)}\right) \\ $$$$=−\sum_{{n}=\mathrm{1}} ^{{N}} \:\:\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left(\frac{{n}\:+{n}+\mathrm{1}}{\mathrm{1}−{n}\left({n}+\mathrm{1}\right)}\right)\:=−\sum_{{n}=\mathrm{1}} ^{{N}} \:\left(−\mathrm{1}\right)^{{n}} \left\{{arctan}\left({n}\right)\:+{arctan}\left({n}+\mathrm{1}\right)\right\} \\ $$$$\Rightarrow−{S}_{{n}} =−{arctan}\left(\mathrm{1}\right)\:−{arctan}\left(\mathrm{2}\right)\:+{arctan}\left(\mathrm{2}\right)\:+{arctan}\left(\mathrm{3}\right)−… \\ $$$$+\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left({n}\right)\:+\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left({n}+\mathrm{1}\right) \\ $$$$=\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left({n}+\mathrm{1}\right)\:−\frac{\pi}{\mathrm{4}}\:\Rightarrow\:{S}_{{N}} =\frac{\pi}{\mathrm{4}}\:−\left(−\mathrm{1}\right)^{{n}} \:{arctan}\left({n}+\mathrm{1}\right)\:. \\ $$
Commented by maxmathsup by imad last updated on 13/Apr/19
$${S}_{{N}} =\frac{\pi}{\mathrm{4}}−\left(−\mathrm{1}\right)^{{N}} \:{arctan}\left({N}+\mathrm{1}\right)\:. \\ $$