1. ## Uniform convergence

Decide if they uniformly convergent:

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

Any help would be appreciated!

2. Originally Posted by doug
Decide if they uniformly convergent:

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

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.

3. Originally Posted by doug
Decide if they uniformly convergent:

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

Any help would be appreciated!
I hope I am not overstepping my boundaries and taking $\displaystyle \varnothing$'s hint further by considering the M-test