Radius of convergence

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January 22nd 2009, 11:09 AM
Haris

Radius of convergence

What would be the radius of convergence of $e^{sinx}$ ?
January 22nd 2009, 11:18 AM
Mathstud28

Quote:

Originally Posted by Haris

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?
January 22nd 2009, 11:20 AM
Haris

Aye, sorry thats the question I wanted to ask.
January 22nd 2009, 11:25 AM
Mathstud28

Quote:

Originally Posted by Haris

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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