1. ## Radius of Convergence / Taylor Series

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?

2. Originally Posted by 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?
Yes that is correct.

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