Use the ratio test to show that the series the summation from n equals 1 to infinity of the quotient of 4 to the nth power and n factorial 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 / a_n = 4ⁿ⁺¹ / (n+1)! / 4ⁿ/n! = 4ⁿ⁺¹ / (n+1)! · n! / 4ⁿ = 4 / (n+1)

lim_n→∞ 4 / (n+1) = 0 < 1 and therefore the sequence converges.