Complex Series, Region of Convergence

In summary, the electrical engineering student used the ratio test to solve a series region of convergence problem. The resulting limit was |z+5i|, which must be smaller than a certain value for the series to converge.
  • #1
sgonzalez90
6
0
Good evening, I'm an electrical engineering student questioning my answer to this series Region of Convergence problem.

Ʃ(0,inf) (n(n-1)(z+5i)^n)/n

Using the ratio test lim n-> |an+1/an|

I was able to get it down to lim n->|n(z+5i)/(n-1)| which gave |inf/inf| = 1, which means the test fails. What do I do to properly solve this? :(
 
Physics news on Phys.org
  • #2
you can't just write it as inf/inf and cancel them!
[tex] \lim_{n\to \infty} |\frac{n(z+5i)}{n-1}| = |z+5i| \lim_{n\to \infty} \frac {n}{n-1} [/tex]
 
  • #3
ROC
= |z+5i| < inf?

:D
 
  • #4
Once you know the limit is |z+5i|, what does this have to be smaller than for the series to converge?
 

1. What is a complex series?

A complex series is an infinite sum of complex numbers. It can be written in the form of a0 + a1z + a2z2 + a3z3 + ..., where an are complex coefficients and z is a complex variable.

2. What is the region of convergence of a complex series?

The region of convergence (ROC) of a complex series is the set of values for the complex variable z for which the series converges. It is typically represented as a circle, annulus, or half-plane in the complex plane.

3. How do I determine the region of convergence of a complex series?

The region of convergence can be determined by using various convergence tests, such as the ratio test or the root test. These tests involve taking the limit of the absolute value of the terms in the series, and the resulting value will determine the shape and size of the region of convergence.

4. Can a complex series have multiple regions of convergence?

Yes, a complex series can have multiple regions of convergence. This can occur when the series has different convergence behavior for different values of z. In this case, the series may converge in one region and diverge in another.

5. What is the significance of the region of convergence in complex analysis?

The region of convergence is important because it tells us where a complex series is valid and where it is not. It also allows us to manipulate the series algebraically within its region of convergence, and to extend the function represented by the series beyond its original domain. Additionally, the region of convergence can provide insight into the behavior of the function and its singularities.

Similar threads

Replies
6
Views
528
  • Calculus and Beyond Homework Help
Replies
2
Views
89
Replies
2
Views
689
  • Calculus and Beyond Homework Help
Replies
5
Views
939
  • Calculus
Replies
3
Views
1K
Replies
3
Views
886
  • Calculus
Replies
19
Views
1K
Replies
2
Views
1K
Replies
37
Views
3K
Replies
8
Views
2K
Back
Top