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## 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.