Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Test the following for uniform convergence:

(a) The sequence of functions {$x^n/(1 + x^n)$} over the interval [0, 2].
(b) The series $ ∑_{n=1}^∞\sin nx/(n^2+ 1)$over $\mathbb{R}$.
(c) The sequence of functions{$n^2x^2e^{−nx}$} over the interval (0, ∞).

How can I solve this problem?

share|improve this question
5  
What have you tried? –  emka Jan 14 '13 at 17:23
    
Which one - or you want generic hints common to all three? –  Jan Dvorak Jan 14 '13 at 17:32
1  
(a) Calculate the pointwise limit and think about the result while keeping in mind that a uniform limit of continuous functions must be continuous. (b) Use the Weierstrass $M$-test. (c) Calculate the pointwise limit, and calculate $f_n(1/n)$, where $f_n(x)=n^2x^2e^{-nx}$. Think about the results... –  David Mitra Jan 14 '13 at 17:52
add comment

1 Answer

These are just short, potentially useful observations.

(a) Notice that $\frac{x^n}{x^n+1} \leq 1$ for $x \in [0,1]$

(b) $\sum\frac{\sin(nx)}{n^2+1}\leq \sum \frac{1}{n^2+1}\leq \sum \frac{1}{n^2}...$

(c) This converges uniformly, but I have to think about it more.

share|improve this answer
1  
For c), note $f_n(x)=n^2x^2e^{-nx}$ has value $1/e$ for $x=1/n$; while the pointwise limit of $(f_n)$ is the zero function on $(0,\infty)$. –  David Mitra Jan 14 '13 at 17:44
    
Thanks. As I noted, I don't know if my answer is particularly useful. I've had these problems before and the information I provided for (a) and (b) helped me think about convergence and uniform convergence. –  emka Jan 14 '13 at 21:32
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.