Please help and provide a detailed solution Find the radius

Please help and provide a detailed solution.

Find the radius and interval of convergence for the power series

Thanks!

Solution

Using ratio test x^(n+1)/(n+3) *((n+2)/x^n We end up with x as (n+2)/(n+3) goes to 1 when x gets large. So radius of convergence is 1 since under ratio test it converges where |x|<1 Testing endpoints it converges by alternating series test when x=-1, diverges by integral test when x=1. So interval of convergence is [-1,1)