Use the ratio test to show that the series is convergent. Be sure to show your work.

Question

 Use the ratio test to show that the series  is convergent. Be sure to show your work.

Answer 1

a_n+1 = 4ⁿ⁺¹ / (n+1)! , a_n = 4ⁿ / n!

a_n+1 / a_n = [4ⁿ⁺¹ / (n+1)! ] / [4ⁿ / n!]

= (4ⁿ⁺¹ / (n+1)!) · (n! / 4ⁿ)

= 4 / (n+1)

lim_n→∞ [4 / (n+1)] = 0 so the series converges.