# Thread: Series-Interval of Convergence-absolutely or conditionally

1. ## Series-Interval of Convergence-absolutely or conditionally

Hi everyone I just had a general question. I know that one way to find the interval of convergence for a power series is through the ratio test. If your answer for the test is less then 1 then the series is absolutely convergent. Then you have to check the end points to see if they converge. My question is the following can the endpoints converge conditionally at the endpoints, while in between the endpoints the series converges absolutely? And can you see this just by using the Alternate Series Test on the endpoints after doing the Ratio Test?

2. Originally Posted by Mente
Hi everyone I just had a general question. I know that one way to find the interval of convergence for a power series is through the ratio test. If your answer for the test is less then 1 then the series is absolutely convergent. Then you have to check the end points to see if they converge. My question is the following can the endpoints converge conditionally at the endpoints, while in between the endpoints the series converges absolutely? And can you see this just by using the Alternate Series Test on the endpoints after doing the Ratio Test?
Typically when you test the endpoints, one of three possibilities happen:

1) The series converges at both endpoints, e.g. $\sum_{n=1}^\infty \frac{x^n}{n^2}$ so here $-1 \le x \le 1$

2) The series converges at only one endpoint, e.g. $\sum_{n=1}^\infty \frac{x^n}{n}$ so here $-1 \le x < 1$

3) The series converges at neither endpoints, e.g. $\sum_{n=1}^\infty x^n$ so here $-1 < x < 1$.