let-v-n-a-1-n-n-1-a-x-2-arctan-1-a-x-dx-with-a-gt-0-1-determine-a-explicit-form-of-v-n-a-2-study-the-convergence-of-n-v-n-a-3-calculate-v-n-1-and-n-v-n-1-

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Question Number 47065 by maxmathsup by imad last updated on 04/Nov/18

let v_n (a)= ∫_(1/n) ^n (1−(a/x^2 ))arctan(1+(a/x))dx with a>0 1) determine a explicit form of v_n (a) 2) study the convergence of Σ_n v_n (a) 3)calculate v_n (1) and Σ_n v_n (1) .

$${let}\:{v}_{{n}} \left({a}\right)=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\left(\mathrm{1}−\frac{{a}}{{x}^{\mathrm{2}} }\right){arctan}\left(\mathrm{1}+\frac{{a}}{{x}}\right){dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{v}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}} \:{v}_{{n}} \left({a}\right) \\ $$$$\left.\mathrm{3}\right){calculate}\:{v}_{{n}} \left(\mathrm{1}\right)\:\:{and}\:\sum_{{n}} {v}_{{n}} \left(\mathrm{1}\right)\:. \\ $$

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let-v-n-a-1-n-n-1-a-x-2-arctan-1-a-x-dx-with-a-gt-0-1-determine-a-explicit-form-of-v-n-a-2-study-the-convergence-of-n-v-n-a-3-calculate-v-n-1-and-n-v-n-1-

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