# Find the interval of convergence

1. Dec 10, 2015

### acdurbin953

1. The problem statement, all variables and given/known data
Find the interval of convergence of the power series ∑(x-2)n / 3n

2. Relevant equations
ρn = |an+1| / |an|

3. The attempt at a solution
I got that ρn = | (x-2) / 3 |. I set my ρn ≤ 1, since this is when the series would be convergent. Manipulating that expression, I got that the interval of convergence is -5 ≤ x ≤ 5. The answer in the back of the book is -1 ≤ x ≤ 5. I am confused where the -1 comes from.

2. Dec 10, 2015

### acdurbin953

I'm sorry, I realized after this post that I was getting interval of convergence mixed up with radius of convergence.

3. Dec 10, 2015

### Ray Vickson

Be careful: the endpoints are generally not included, because they would correspond to a series of the form $\sum 1^n$ or $\sum (-1)^n$. Are you sure the book did not say $-1 < x < 5$?

4. Dec 13, 2015

### acdurbin953

You're right, it does say -1 < x < 5. Thank you for that reminder!