MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the series n=1∑∞n(−1)n. Which statement best explains why this series is conditionally convergent but not absolutely convergent?
(a) Because the ratio test gives a limit of 1, the series converges but not absolutely.
(b) The terms decrease to 0, so the alternating series test guarantees convergence, but ∑1/n diverges, so the series is not absolutely convergent.
(c) Applying the root test to ∣an∣ yields a limit less than 1, proving absolute convergence.
(d) Since ∑1/n is a p-series with p=21>1, it converges; hence the given series converges absolutely.