let-give-I-n-0-dt-1-t-2-n-with-n-integr-and-n-1-1-prove-the-convergence-of-I-n-2-find-lim-n-I-n-3-study-the-convergence-of-the-serie-n-1-1-n-I-n-

Tinku Tara June 4, 2023 Relation and Functions 0 Comments

Question Number 31966 by abdo imad last updated on 17/Mar/18

let give I_n = ∫_0 ^∞ (dt/((1+t^2 )^n )) with n integr and n≥1 1) prove the convergence of I_n 2)find lim_(n→∞) I_n 3) study the convergence of the serie Σ_(n=1) ^∞ (−1)^n I_n .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{{n}} }\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{convergence}\:{of}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow\infty} \:\:{I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:{the}\:{serie}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\:{I}_{{n}} \:. \\ $$

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