Convergent or Divergent?

Question

Match each of the following with the correct statement.

A. The series is absolutely convergent.

C. The series converges, but is not absolutely convergent.

D. The series diverges.

a) sigma (n=1 to inf) [(-1)^n 2^(n-1)]/[2^(n+1) n^(1/7)

b) sigma (n=1 to inf) n^4/7^n

c) sigma (n=1 to inf) (n+2)!/[(3^n)(n!)]

d) sigma (n=1 to inf) (-1)^n (n!)/ 5^n

e) sigma (n=1 to inf) (-1)^n / [6^n n!)

Thank you!

Answer 1

Since the first one reduces to the sum of [(-1)^n]/[4 n^(1/7)] this is divergent by comparison with p-series. (If a series fails to converge by the alternating series test and its absolute series diverges, then the series is divergent, if I'm not mistaken - - this series fails the test for convergence by alternating series and its corresponding absolute value series fails by p-series comparison.)