Using the Weierstrass M-test to check for uniform convergence

[nietzsche]

Homework Statement

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

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 x \in [1,\infty) we have

\frac{nx}{1+n^4 x^2} \leq \frac{n}{n^4} = \frac{1}{n^3}

and \sum_{n=1}^\infty {\frac{1}{n^3}} 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.

 

[Eynstone]

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.