Radius of Convergence / Taylor Series

May 2010
10
0
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

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Feb 2008
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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}
 
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