Determine the radius and interval of convergence of .

Question

Determine the radius and interval of convergence of  .

Quizplus · Accepted Answer

We can apply the ratio test to determine the radius of convergence:

|a_{n+1}/a_n| = |(11(x+1))/(n+1)(n+2) * n(n-1)/11x|
= |(x+1)/x| * (n/(n+2))

The sequence (n/(n+2)) converges to 1 as n approaches infinity, therefore:

|a_{n+1}/a_n| = |(x+1)/x| * (n/(n+2)) → |(x+1)/x|

The series converges for |(x+1)/x|<1, or equivalently, for |x+1|<|x|. This gives -2<x<0 as the interval of convergence.

So the answer is C, with radius of convergence 2 and interval of convergence (-2,0).

Determine the Radius and Interval of Convergence of