Use the alternating series test to determine the convergence/divergence of the series

Question

 

Use the alternating series test to determine the convergence/divergence of the series  .

A	The alternating series fails.
B	The series converges.
C	The series is conditionally convergent.
D	The summation must start with n = 0 so the alternating series test cannot be applied.

Answer 1

This series converges conditionally: Series itself converges by alternative series test, but Ia_nI = n²/(n³ + 1).

This series diverges by limit comparison test: lim [n²/(n³ + 1)]/(1/n) = 1 and because series b_n = 1/n

n → ∞

diverges so series a_n = n²/(n³ + 1) diverges. Because given series conditionally convergent