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Math Help - Uniform convergence

  1. #1
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    Uniform convergence

    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!
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  2. #2
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    Quote Originally Posted by doug View Post
    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.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by doug View Post
    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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