Complex series: Circle of convergence

In summary, the conversation discusses the use of the ratio test for complex power series and its application in finding the circle of convergence. It is stated that the ratio test gives the same answer regardless of the substitution of variables and can work for power series starting at any value of n. Additionally, it is mentioned that the limiting behavior is only affected near the limit, so the starting point of the series does not matter.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

Lets say I have a complex power series given by

[tex]
\sum_{n=0}^\infty c_nz^n,
[/tex]

where z is a complex number and c is a complex constant. Inside its circle of convergence, I can differentiate it leading to

[tex]
\sum_{n=0}^\infty c_nnz^{n-1} = \sum_{n=0}^\infty (n+1)c_{n+1}nz^{n}.
[/tex]

If I want to find the circle of convergence for this series, then I can use the ratio test

[tex]
\frac{1}{R} = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right|.
[/tex]

I have two questions for this:

Question #1: Does the ratio test give me the same answer regardless of I substitute e.g. n -> n+3? I.e., is it correct that

[tex]
\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 3} }}{{a_{n + 2} }}} \right|.
[/tex]

Personally, I think it does not matter, because we let n go to infinity.

Question #2: Does the ratio test only work for power series that go from n=0 to infinity, or do they also work if n start at e.g. 1 or -1?Thank you very much in advance.Niles.
 
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  • #2
Well the ratio test is typically derived from comparison to a geometric series. Moreover, one way to derive the comparison test is based on the monotone convergence theorem applied to partial sums. So I think you're right on both counts since only long term behavior matters. Especially for question 2, remember that including or excluding a finite number of terms in an infinite series does not affect convergence.
 
  • #3
Hi Niles! :smile:
Niles said:
Question #1: Does the ratio test give me the same answer regardless of I substitute e.g. n -> n+3? I.e., is it correct that

[tex]
\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 3} }}{{a_{n + 2} }}} \right|.
[/tex]

Personally, I think it does not matter, because we let n go to infinity.

Question #2: Does the ratio test only work for power series that go from n=0 to infinity, or do they also work if n start at e.g. 1 or -1?

Your intuitive suspicions are completely correct …

limiting behaviour is only affected "near" the limit …

what happens at the other end doesn't matter!

So yes, to both #1 and #2. :smile:
 

1. What is a complex series?

A complex series is a sequence of complex numbers that are added together. It can be represented in the form of ∑(a_n), where a_n is the n-th term in the sequence.

2. What is a circle of convergence in a complex series?

A circle of convergence is a boundary within which the series will converge. It is determined by the values of the complex numbers in the series and can be represented by a circle on the complex plane.

3. How is the circle of convergence determined?

The circle of convergence is determined by applying the ratio test to the series. This test compares the ratio of consecutive terms in the series to a limit, and if the limit is less than 1, the series will converge within the circle of convergence.

4. What happens if a complex series does not converge within the circle of convergence?

If a complex series does not converge within the circle of convergence, it will either diverge or have a different circle of convergence. This can happen if the limit in the ratio test is greater than 1, indicating that the series will not converge.

5. Can the circle of convergence change for a complex series?

Yes, the circle of convergence can change for a complex series. It can change if the values of the complex numbers in the series change, or if a different method is used to determine the circle of convergence, such as the root test or the integral test.

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