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Math Help - Uniform Convergence of Power Series

  1. #1
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    Uniform Convergence of Power Series

    Suppose \sum_{n=0}^{\infty} {a_n}x^n converges on [-R, R] Suppose \delta < R. Show that \sum_{n=0}^{\infty} {a_n}x^n converges uniformly on [-\delta, \delta].

    Hint given: Pick r such that \delta < r < R. You are given that \sum_{n=0}^{\infty} {a_n}r^n converges, therefore |a_n|r^n \rightarrow 0. In particular, |a_n|r^n \leq C, where C is independent of n.

    I am unable to show that {a_n}r^n converges absolutely, while the hint suggests that I should show this. Thanks in advance for any help!
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  2. #2
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    Quote Originally Posted by h2osprey View Post
    Suppose \sum_{n=0}^{\infty} {a_n}x^n converges on [-R, R] Suppose \delta < R. Show that \sum_{n=0}^{\infty} {a_n}x^n converges uniformly on [-\delta, \delta].

    Hint given: Pick r such that \delta < r < R. You are given that \sum_{n=0}^{\infty} {a_n}r^n converges, therefore |a_n|r^n \rightarrow 0. In particular, |a_n|r^n \leq C, where C is independent of n.

    I am unable to show that {a_n}r^n converges absolutely, while the hint suggests that I should show this. Thanks in advance for any help!
    I can see nothing here that suggests absolute convergence. |a_n|r^n is NOT |a_nr^n|.
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