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How to show that $\sum_{n=1}^{\infty}\frac{x^{2n}}{(x+n)^{3/2}}$ is uniformly continuous on $[0,1]$

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Hint: Show that $\frac{x^{2n}}{(x+n)^{3/2}}\leq \frac 1{n^{3/2}}$, hence the series is _ (fill in the blank) convergent. What about the sum of a series of uniformly continuous functions which is __ (fill in the blank) convergent?

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    ............. :-)2012-04-01
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WolframAlpha says:

$\sum_{n=1}^{\infty}\frac{x^{2n}}{(x+n)^{3/2}}= x^2 \Phi\left( x^2,\frac{3}{2},x+1 \right)$, when $|x|<1$. $\Phi(z,s,a)$ gives the Hurwitz Lerch transcendent.