# Radius of Convergence / Taylor Series

#### fourierwarrior

Hello everyone! I'm new here, this is my first post. I'm currently practicing some past examination papers, and I'm stuck on the following question:

"Find the Taylor series of the function $$\displaystyle e^z$$ about the point z = 2, and state the radius of convergence of the power series."

So far, I've found the Taylor series to be $$\displaystyle \sum{e^2(z-2)^n/n!}$$ (anybody know how to include the from n=0 to infinity part?)

I've looked at the radius of convergence pages on wikipedia and wolfram, but I'm a little shakey when it comes to understanding how to use the ratio test.

As stated on wikipidea (Ratio test - Wikipedia, the free encyclopedia) After working out $$\displaystyle \frac{e^2(z-2)^{n+1}/(n+1)!}{e^2(z-2)^n/n!}$$, I got $$\displaystyle \frac{z-2}{(n+1)}$$, so would I be correct in saying that the series converges for all z, since the limit of this function tends to zero as n tends to infinity?

#### TheEmptySet

MHF Hall of Honor
Hello everyone! I'm new here, this is my first post. I'm currently practicing some past examination papers, and I'm stuck on the following question:

"Find the Taylor series of the function $$\displaystyle e^z$$ about the point z = 2, and state the radius of convergence of the power series."

So far, I've found the Taylor series to be $$\displaystyle \sum{e^2(z-2)^n/n!}$$ (anybody know how to include the from n=0 to infinity part?)

I've looked at the radius of convergence pages on wikipedia and wolfram, but I'm a little shakey when it comes to understanding how to use the ratio test.

As stated on wikipidea (Ratio test - Wikipedia, the free encyclopedia) After working out $$\displaystyle \frac{e^2(z-2)^{n+1}/(n+1)!}{e^2(z-2)^n/n!}$$, I got $$\displaystyle \frac{z-2}{(n+1)}$$, so would I be correct in saying that the series converges for all z, since the limit of this function tends to zero as n tends to infinity?
Yes that is correct.

The code you are looking for is \sum_{n=0}^{\infty}

• fourierwarrior