Question

integrate x^n/(1+x^2)^m limit o to infinity

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Answer 1

Answer:

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If x=0 or x=1 it is trivial, so 0<x<1. Define a=1−x2, then 0<a<1.

limn→∞n2x(1−x2)n=xlimn→∞n2an≤limn→∞n2an=limn→∞n2(1/a)n=limn→∞2nln(1/a)(1/a)n=limn→∞2ln2(1/a)(1/a)n=2ln2(1/a)limn→∞an=0

Using l'Hopital twice and the fact that 0<a<1.

Also all of the terms in the sequence are positive, 0≤limn→∞n2x(1−x2)n≤0.