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Thread: Uniform Convergence of Power Series

  1. #1
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    Uniform Convergence of Power Series

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

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

    I am unable to show that $\displaystyle {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 $\displaystyle \sum_{n=0}^{\infty} {a_n}x^n $ converges on $\displaystyle [-R, R]$ Suppose $\displaystyle \delta < R. $ Show that $\displaystyle \sum_{n=0}^{\infty} {a_n}x^n $ converges uniformly on $\displaystyle [-\delta, \delta]$.

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

    I am unable to show that $\displaystyle {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. $\displaystyle |a_n|r^n$ is NOT $\displaystyle |a_nr^n|$.
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