Complex series: Circle of convergence

by Niles
Tags: circle, complex, convergence, series
 P: 1,863 1. The problem statement, all variables and given/known data Hi all. Lets say I have a complex power series given by $$\sum_{n=0}^\infty c_nz^n,$$ where z is a complex number and c is a complex constant. Inside its circle of convergence, I can differentiate it leading to $$\sum_{n=0}^\infty c_nnz^{n-1} = \sum_{n=0}^\infty (n+1)c_{n+1}nz^{n}.$$ If I want to find the circle of convergence for this series, then I can use the ratio test $$\frac{1}{R} = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right|.$$ 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 $$\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 3} }}{{a_{n + 2} }}} \right|.$$ 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. Best regards, Niles.
 P: 1,104 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.
HW Helper
Thanks
P: 26,148
Hi Niles!
 Quote by Niles 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 $$\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 3} }}{{a_{n + 2} }}} \right|.$$ 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.

 Related Discussions Calculus & Beyond Homework 2 General Math 0 Precalculus Mathematics Homework 28 Calculus 0 Calculus & Beyond Homework 29