Find the interval of convergence

[acdurbin953]

Homework Statement

Find the interval of convergence of the power series ∑(x-2)ⁿ / 3ⁿ

Homework Equations

ρ_n = |a_n+1| / |a_n|

The Attempt at a Solution

I got that ρ_n = | (x-2) / 3 |. I set my ρ_n ≤ 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.

 

[acdurbin953]

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

 

[Ray Vickson]

Homework Statement

Find the interval of convergence of the power series ∑(x-2)ⁿ / 3ⁿ

Homework Equations

ρ_n = |a_n+1| / |a_n|

The Attempt at a Solution

I got that ρ_n = | (x-2) / 3 |. I set my ρ_n ≤ 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.

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 $-1 < x < 5$?

 

[acdurbin953]

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 $-1 < x < 5$?

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