(adsbygoogle = window.adsbygoogle || []).push({}); (Problem 64 from practice math subject GRE exam:)For each positive integer n, let [tex]f_n[/tex] be the function defined on the interval [0,1] by [tex]f_n(x)=\frac{x^n}{1+x^n}[/tex]. Which of the following statements are true?

I. The sequence [tex]\{f_n\}[/tex] converges pointwise on [0,1] to a limit function f.

II. The sequence [tex]\{f_n\}[/tex] converges uniformly on [0,1] to a limit function f.

III. [tex]\lim_{n\rightarrow\infty}\int_0^1f_n(x)dx=\int_0^1\lim_{n\rightarrow\infty}f_n(x)dx[/tex]

I believe the sequence does converge pointwise since [tex]f_n(x)\rightarrow0[/tex] when for [tex]x\in\[0,1)[/tex] and [tex]f_n(1)=\frac{1}{2}[/tex] for all n. So the sequence converges to the function f(x)=0 for x < 1 and f(x)=1/2 for x=1.

I'm not too familiar with uniform convergence - looked it up online. Is it enough to say that the sequence does not converge uniformly because the limit function f is discontinuous?

I don't know how to prove the last one ... it seems quite obvious to me (that you could interchange order of the limit and the integral).In what situations would this not be allowed and how can I check if, in this specific case, I can?

Thanks!

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# Homework Help: Pointwise v. uniform convergence

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