• Support PF! Buy your school textbooks, materials and every day products Here!

Using the Weierstrass M-test to check for uniform convergence

  • Thread starter nietzsche
  • Start date
  • #1
186
0

Homework Statement



[tex]\text{Show that the series }f(x) = \sum_{n=0}^\infty{\frac{nx}{1+n^4 x^2}}\text{ converges uniformly on }[1,\infty].[/tex]

Homework Equations





The Attempt at a Solution



I think we should use the Weierstrass M-test, but I'm not sure if I'm applying it correctly:

For [tex]x \in [1,\infty)[/tex] we have

[tex]\frac{nx}{1+n^4 x^2} \leq \frac{n}{n^4} = \frac{1}{n^3}[/tex]

and [tex]\sum_{n=1}^\infty {\frac{1}{n^3}}[/tex] is a convergent p-series.

Also, for n=0, we have the series function is equal to 0, so this will not affect the convergence of the series.

Therefore, by the Weierstrass M-test, the original series is uniformly convergent.

Have I got it right? Thanks in advance.
 

Answers and Replies

  • #2
336
0
Your estimate is right : nx/(1 + n^4* x^2) attains a maxima equal to 1/2n
(at x = 1/n^2) & decreases there onwards.
 

Related Threads for: Using the Weierstrass M-test to check for uniform convergence

Replies
5
Views
6K
Replies
56
Views
6K
Replies
2
Views
2K
Replies
0
Views
3K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
3
Views
3K
Top