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Math Help - Functional Series and Uniform Convergence Question

  1. #1
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    Functional Series and Uniform Convergence Question

    Let a \in (0,1). Show that the functional series

    \displaystyle\sum_{j=0}^{\infty}(-t^2)^j where t \in [-a,a]

    is uniformly convergent with the limit function

    f(t) = \frac{1}{1+t^2}

    I have absolutely no clue where to go from here.
    Any help???
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by garunas View Post
    Let a \in (0,1). Show that the functional series

    \displaystyle\sum_{j=0}^{\infty}(-t^2)^j where t \in [-a,a]

    is uniformly convergent with the limit function

    f(t) = \frac{1}{1+t^2}

    I have absolutely no clue where to go from here.
    Any help???
    Merely note that |f_j(t)|=t^{2j}\leqslant (a)^{2j} for t\in[-a,a], but since \displaystyle \sum_{j=}^{\infty}a^{2j} converges, it follows by the Weierstrass M-test that \displaystyle \sum_{j=0}^{\infty}f_j(t) converges uniformly on [-a,a]. To prove what it sums to merely note that \displaystyle \frac{1}{1-\left(-t^2\right)}=\sum_{j=0}^{\infty}\left(-t^2\right)^j
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