Find Radius of Convergence of this Complex Series

In summary, the conversation is about finding the radius of convergence for the complex series (Sigma n=1 to infinity) (z - e)^n! and how to obtain it using the ratio test. The answer is R=1 and the reasoning behind it is that when |z-e| < 1, the expression becomes very small and when |z-e| > 1, it becomes very large. This is determined by the large exponent in the expression.
  • #1
Ad123q
19
0
Hi, am a bit stuck with this.

Find the radius of convergence of the complex series

(Sigma n=1 to infinity) (z - e)^n!

I know that the answer is R=1 but I'm not sure how to get there.
It's the factorial as a power which I'm not sure about, have seen this in some other problems too.
I have tried using the Ratio Test to determine radius of convergence but doesn't seem to be working.

Any ideas anyone?

Thanks in advance, and apologies for the lack of Latex.
 
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  • #2
The ratio test should work. The ratio of the n term to the (n-1) term is (z-e)n.
If |z-e| < 1, the ratio -> 0, while if |z-e| > 1, tie ratio becomes infinite.
 
  • #3
Thanks, though I'm still not 100% sure about how to get this.

Using ratio test I obtain

mod[ (z-e)^(n+1)! / (z-e)^n! ]
= mod[ (z-e)^n!(n+1-1) ]
= mod[ (z-e)^n.n! ]

Then I'm not sure where to go from here.
 
  • #4
mod[ (z-e)^n.n! ] = [mod(z-e)]^n.n!

When mod(z-e) < 1, the large exponent makes the expression very small, while if mod(x-e) > 1, it makes the expression very large. Therefore the radius of convergence = 1.
 
Last edited:

FAQ: Find Radius of Convergence of this Complex Series

What is the general formula for finding the radius of convergence of a complex series?

The general formula for finding the radius of convergence of a complex series is given by the ratio test, which states that the radius of convergence is equal to the limit of the absolute value of the ratio of consecutive terms in the series as the index approaches infinity.

How do you apply the ratio test to determine the radius of convergence?

To apply the ratio test, you first take the absolute value of the ratio of consecutive terms in the series. Then, you take the limit of this ratio as the index approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive and another test must be used.

Can the radius of convergence be negative?

No, the radius of convergence is always a positive number. It represents the distance from the center of the complex series to the nearest point where the series converges.

Is the radius of convergence affected by the choice of the center of the complex series?

Yes, the radius of convergence is affected by the choice of the center of the complex series. The closer the center is to the point where the series converges, the larger the radius of convergence will be.

What happens if the ratio test is inconclusive?

If the ratio test is inconclusive, it means that the radius of convergence cannot be determined using this test alone. In this case, another test such as the root test or the alternating series test may be used to determine the radius of convergence.

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