Uniform convergence

**doug** · March 2nd 2011, 10:28 AM

Decide if they uniformly convergent:

$\sum_{n=1}^{\infty}\frac{x^2}{1+n^2x^2}$ on [0,1]
$\sum_{n=1}^{\infty}\frac{1}{n^2}e^{-nx^2}$ on $\mathbb R$

Any help would be appreciated!

**TheEmptySet** · March 2nd 2011, 11:31 AM

Originally Posted by doug

Decide if they uniformly convergent:

$\sum_{n=1}^{\infty}\frac{x^2}{1+n^2x^2}$ on [0,1]
$\sum_{n=1}^{\infty}\frac{1}{n^2}e^{-nx^2}$ on $\mathbb R$

Any help would be appreciated!

Hint: Where does the maximum of each function occur. Use that x value to find a series that dominates the above. Is this new series uniformly convergent? If so what can you conclude.

**Drexel28** · March 2nd 2011, 06:42 PM

Originally Posted by doug

Decide if they uniformly convergent:

$\sum_{n=1}^{\infty}\frac{x^2}{1+n^2x^2}$ on [0,1]
$\sum_{n=1}^{\infty}\frac{1}{n^2}e^{-nx^2}$ on $\mathbb R$

Any help would be appreciated!

I hope I am not overstepping my boundaries and taking $\varnothing$ 's hint further by considering the M-test

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