Question Number 47065 by maxmathsup by imad last updated on 04/Nov/18
$${let}\:{v}_{{n}} \left({a}\right)=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\left(\mathrm{1}−\frac{{a}}{{x}^{\mathrm{2}} }\right){arctan}\left(\mathrm{1}+\frac{{a}}{{x}}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{v}_{{n}} \left({a}\right) \\ $$ $$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}} \:{v}_{{n}} \left({a}\right) \\ $$ $$\left.\mathrm{3}\right){calculate}\:{v}_{{n}} \left(\mathrm{1}\right)\:\:{and}\:\sum_{{n}} {v}_{{n}} \left(\mathrm{1}\right)\:. \\ $$