# Uniform convergence in a neighborhood of zero.

This is about a problem from Carlslaw's book on Fourier series (the 1929 edition).

One has to show that following series is uniformly convergent for "all values of $x$" ( I assume this means one has to show the series is uniformly convergent on $\left[0,\infty\right)$)

$$\sum_{n=1}^{\infty} \frac{x^{3/2}}{1+n^2x^2}.$$

One can easily verify the uniform convergence on any $[X,\infty)$ where $X>0$ using the M-test, since for any $x \geq X$ we have $$0 \leq \frac{x^{3/2}}{1+n^2x^2} = \frac{1}{ \sqrt{x} \times \left(\frac{1}{x^2} + n^2\right)} \leq \frac{1}{\sqrt{X}n^2}.$$

It is also easy to check the continuity of the above series at $0$ since for any $x >0$ we have

$$0 \leq \sum_{n=1}^{\infty} \frac{x^{3/2}}{1+n^2x^2} \leq x^{3/2} \times \int_{0}^{\infty}\dfrac{dt}{1+t^2x^2} = \sqrt{x} \frac{\pi}{2}.$$

So the uniform continuity in a neighborhood of $0$ appears plausible, however I am unable to show this.

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We have $$\sum_{n=1}^\infty\frac{x^{3/2}}{1+n^2x^2} =\sum_{n=1}^\infty\frac1{n^{3/2}}\frac{n^{3/2}x^{3/2}}{1+n^2x^2}$$ and since for all $u\ge0$, $$\frac{u^{3/2}}{1+u^2}\le\frac{3^{3/4}}{4}$$ and the sum of $n^{-3/2}$ converges, we get that the series converges uniformly,