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Math Help - Radius of convergence

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    Radius of convergence

    What would be the radius of convergence of e^{sinx}?
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Haris View Post
    What would be the radius of convergence of e^{sinx}?
    This is not a question? What does this mean?

    What is the ROC of \sum_{n=0}^{\infty}e^{n\sin(x)} maybe?
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    Aye, sorry thats the question I wanted to ask.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Haris View Post
    Aye, sorry thats the question I wanted to ask.
    This is just a geometric series, so we need the absolute value of the ratio, e^{\sin(x)} in this case, to be less than one.

    So this series converges for all values of x in the set S=\left\{x:|e^{\sin(x)}|=e^{\sin(x)}<1\right\}

    Note that we have that 0<e^x<1 iff x<0. So in our case we need \sin(x)<0 or that \pi<x<2\pi or 3\pi<x<4\pi or in general (2n-1)\pi<x<2n\pi~~n\in\mathbb{Z}
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