Which of the following is the radius?

Question

Which of the following is the radius of convergence of the series ∑ (n(2x-5)^n)/(n+2) from n=0 to infinity?

 

a) 1

b) 1/2

c) 3/2

d) 2

e) 5/2

 

I worked the problem and the answer I got is 1. Can anyone please verify it? Please show all your work. Thanks.

Answer 1

Sun, here's Paul's online notes for Calculus II:

http://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx

 

The Sequences topic is there.

 

I only taught Calc I plus some of the Integration stuff of Calc II, so it's an area of study for me too.

Answer 2

Hi Sun!

 

Using the ratio test:

 

Using the ratio test lim(n→∞) |((n+1) (2x-5)^(n+1) (n+2))/((n+3)n(2x-5)^n )|=lim(n→∞) |2x-5|.lim(n→∞) |(n+1)(n+2)/(n+3)n|=lim(n→∞) |2x-5|.
The series converge when |2x-5|<1, so x=5/2.