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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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2 Answers 2

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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The title and question don't seem to match :-) –  Aryabhata Apr 1 '12 at 19:58
    
@Aryabhata I agree. –  Davide Giraudo Apr 1 '12 at 19:59
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which is _________ convergent (fill in the blank)? (Using underscore character...) –  Aryabhata Apr 1 '12 at 20:07
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@Aryabhata And all this time I was using "\underline{\phantom{Iwantittobethiswide}}" –  David Mitra Apr 1 '12 at 20:19
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............. :-) –  Aryabhata Apr 1 '12 at 20:19

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.

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