# Radius of convergence

• Jan 22nd 2009, 11:09 AM
Haris
What would be the radius of convergence of $\displaystyle e^{sinx}$?
• Jan 22nd 2009, 11:18 AM
Mathstud28
Quote:

Originally Posted by Haris
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?
• Jan 22nd 2009, 11:20 AM
Haris
Aye, sorry thats the question I wanted to ask.
• Jan 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, $\displaystyle e^{\sin(x)}$ 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 $\displaystyle (2n-1)\pi<x<2n\pi~~n\in\mathbb{Z}$