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I want to show the uniform convergence of the series $$\sum_{n=1}^{\infty} \frac{x}{n(1+nx^2)}$$

It's easy to show the uniform convergence for $|x| > \epsilon$, for any positive $\epsilon$, by using Weierstrass M-test. But I'm having trouble showing the uniform convergence in a neighborhood of zero. Any hints?

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I think this has already been asked. It'd be good if you also gave yourself a nickname or name which could identify you. – Pedro Tamaroff Feb 22 '12 at 19:40
up vote 2 down vote accepted

Let $f_n(x):=\frac x{n(1+nx^2)}$. Then $$f'_n(x)=\frac 1n\frac{1+nx^2-2nxx}{(1+nx^2)^2}=\frac 1n\frac{1-nx^2}{(1+nx^2)^2}$$ so $f_n$ reaches its maximum on $\mathbb R_+$ at $x_n:=\frac 1{\sqrt n}$. We have $f_n(x_n)=\frac{n^{-1/2}}{n(1+n(n^{-1/2})^2)}=\frac 1{2n^{3/2}}$, and since $\sum_{n\geq 1}\frac 1{n^{3/2}}$ is convergent we conclude that the series $\sum_n f_n$ is normally convergent on $\mathbb R_+$, and since each $f_n$ is odd we have the normal hence uniform convergence on $\mathbb R$.

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Wow, that's nice, I've never encountered normal convergence before, but that's very useful. – StuartHa Feb 22 '12 at 19:56

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