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I am trying to show that if $$u_{n}=-2\left( n-1\right) ^{2}x e^{-\left( n-1\right) ^{2}x^{4}}+2n^{2}xe^{-n^{2}x^{2}}$$ then the series $\sum _{n=1}^{\infty }u_{n}\left( x\right)$ does not converge uniformly near $x =0$. Due to complexity in the expression finding a partial sum seems rather daunting, here, so i guess the standard necessary condition for uniformity of convergence seems hard to verify. I also considered Weierstrass's condition for uniform convergence but no convergent series comes to mind such that moduli of terms of this series are less than the corresponding terms in the convergent series.

Any help with a proof strategy would be much appreciated.

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Note that for $s > 0$, $f(s,t) = t e^{-st}$ as a function of $t > 0$ has its maximum at $t=1/s$, where $f(s,1/s) = e^{-1}/s$. Now $u_n(x) = -2 f(x^3, (n-1)^2 x) + 2 f(x, n^2 x)$. If we take $x = 1/\sqrt{n-1}$, we get $$u_n(1/\sqrt{n-1}) = - 2 (n-1)^{3/2} e^{-1} + \frac{2n^2}{\sqrt{n-1}}\ e^{-n^2/(n-1)} $$ and it is easy to see that this goes to $-\infty$ as $n \to \infty$.

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thank you very much, that was very educational, i had n't considered the trick of substituting a value for x, –  Hardy Mar 20 '12 at 19:49

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