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Thread: Radius of convergence

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    Radius of convergence

    What would be the radius of convergence of $\displaystyle e^{sinx}$?
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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 $\displaystyle e^{sinx}$?
    This is not a question? What does this mean?

    What is the ROC of $\displaystyle \sum_{n=0}^{\infty}e^{n\sin(x)}$ 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, $\displaystyle e^{\sin(x)}$ in this case, to be less than one.

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

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