Find the interval of convergence

  • #1

Homework Statement


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

Homework Equations


ρn = |an+1| / |an|

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.
 

Answers and Replies

  • #2
I'm sorry, I realized after this post that I was getting interval of convergence mixed up with radius of convergence.
 
  • #3
Ray Vickson
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Homework Statement


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

Homework Equations


ρn = |an+1| / |an|

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.
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
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!
 

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