Proving Convergence of e^z Series Using Ratio Test on Coefficients

In summary, the conversation discusses proving the convergence of e^z on all complex numbers. The attempt involves using the ratio test on the coefficients of the series and finding the radius of convergence. The formula for e^z is given and the ratio test is explained, as well as its application to finding the values for which the series is convergent. The concept of radius of convergence is discussed and it is determined that when the ratio test of coefficients gives 0, the radius of convergence is infinite.
  • #1
Poirot1
245
0
I am trying to prove e^z converges on all C. Here is my attempt.

e^z=series(z^n/n!)

use the ratio test on the coefficents 1/n! gives lim(1/(n+1))=0, which from rudin means radius of convergence R=1/0 -> R=infinity.
 
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  • #2
Poirot said:
I am trying to prove e^z converges on all C. Here is my attempt.

e^z=series(z^n/n!)

use the ratio test on the coefficents 1/n! gives lim(1/(n+1))=0, which from rudin means radius of convergence R=1/0 -> R=infinity.

You should know that $\displaystyle e^z = \sum_{z = 0}^{\infty}\frac{z^n}{n!}$.

The ratio test states that when you evaluate $\displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|$, if this limit is less than 1, the series is convergent, if this limit is greater than 1, the series is divergent, and if the limit is 1, the test is inconclusive. Since you are trying to find the values for which this series is convergent, you need to set $\displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1$, simplify, and see what values of z will satisfy that inequality.
 
  • #3
Poirot said:
I am trying to prove e^z converges on all C. Here is my attempt.

e^z=series(z^n/n!)

use the ratio test on the coefficents 1/n! gives lim(1/(n+1))=0, which from rudin means radius of convergence R=1/0 -> R=infinity.

The ratio test applies to the terms not the coefficients.

CB
 
  • #4
Yes you can do it that way but you can also do the test on the coefficents then let R be the reciprocal. I have done this before so I know it works, I was just wondering when you get 0 can you just say 1/0 = infinity?
 
  • #5
No, you can't "just say" that. However, it is fairly easy to prove that if $a_n$ converges to 0 then $\frac{1}{a_n}$ does not converge. If you add that $a_n> 0$ for all n, then if diverges to $+\infty$.
 
  • #6
In the context of radius of convergence R must be greater than or equal to 0. I'm pretty convinced that, for power series, whener the ratio/root test of co-efficents gives 0, then we have infinite radius of convergence. Anyone who cares to contradict that is free to give a counter example.
 
  • #7
Poirot said:
In the context of radius of convergence R must be greater than or equal to 0. I'm pretty convinced that, for power series, whener the ratio/root test of co-efficents gives 0, then we have infinite radius of convergence. Anyone who cares to contradict that is free to give a counter example.

Yes you are correct. The ratio test give the radius of convergence whenever the limit exists. (http://en.wikipedia.org/wiki/Power_series#Radius_of_convergence) The root test also give the radius of convergence. (http://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem)
 

FAQ: Proving Convergence of e^z Series Using Ratio Test on Coefficients

What is the ratio test used for in proving convergence of e^z series?

The ratio test is a method for determining whether an infinite series converges or diverges. It involves taking the limit as n approaches infinity of the ratio of the (n+1)th term to the nth term in the series. If this limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it is equal to 1, the test is inconclusive.

How does the ratio test apply specifically to e^z series?

The ratio test can be used on e^z series by examining the coefficients of the series. The general form of an e^z series is Σa_n(z^n)/n!, and the ratio of the (n+1)th coefficient to the nth coefficient can be simplified to just (z/(n+1)). By taking the limit as n approaches infinity, we can determine whether the series converges or diverges.

What is the mathematical formula for the ratio test?

The mathematical formula for the ratio test is lim|a_n+1/a_n| as n→∞. If the limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it is equal to 1, the test is inconclusive.

Can the ratio test be used on all e^z series?

No, the ratio test can only be used on e^z series that have positive coefficients. If the coefficients are negative, the ratio test will not work and a different method must be used to determine convergence or divergence.

What is the advantage of using the ratio test in proving convergence of e^z series?

The ratio test is advantageous because it is a simple and straightforward method for determining convergence or divergence of a series. It also allows for easy comparison between different series and can be used to find the radius of convergence for a power series.

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