- #1

- 64

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**(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!