Consider the series

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑n=1∞(−1)n+11n\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}\dfrac{1}{n}. Which of the following best justifies its convergence? (a) Because the series of absolute values ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty}\frac1n converges, the given alternating series converges absolutely. (b) The magnitudes of the terms decrease monotonically to 00, so by the Alternating Series Test the series converges. (c) Since ∑n=1∞1n\displaystyle\sum_{n=1}^{\infty}\frac1n is a pp-series with p=1p=1, the given series converges by the pp-series test. (d) Because lim⁡n→∞∣1n∣n=0<1\displaystyle\lim_{n\to\infty}\sqrt[n]{\left|\tfrac1n\right|}=0<1, the series converges by the root test.

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