Find all the values of x such that the given series would co

Find all the values of x such that the given series would converge. Give your answer ininterval notation.

Find all the values of x such that the given series would converge. Give your answer ininterval notation. sum_{n=1}^infty frac{(2)^n (x^n)(n+1)}{n+8} Answer:

Solution

Using ratio test 2^(n+1) x^(n+1) (n+2)/(n+9) * (n+8)/2^n x^n (n+1) We are left with |2(x)|< 1 as the other terms cancel or go to 1 as n gets large. So we have (-1/2, 1/2) Testing the endpoints -1/2 and 1/2 do not converge because the terms go to -1 and 1 as n gets large. So (-1/2, 1/2)