Suppose is a sequence of positive terms such that but is not eventually decreasing

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

Problem

Suppose {bn}\{b_n\} is a sequence of positive terms such that lim⁡n→∞bn=0\lim_{n\to\infty} b_n=0 but bnb_n is not eventually decreasing. Consider the alternating series ∑n=1∞(−1)nbn\sum_{n=1}^{\infty} (-1)^{n} b_n. Which of the following best describes what can be concluded about the convergence of the series? (a) The series converges conditionally because lim⁡n→∞bn=0\displaystyle\lim_{n\to\infty} b_n =0. (b) The series diverges since the terms do not decrease monotonically. (c) The standard alternating-series test is inapplicable; additional information or a different test is needed to determine convergence. (d) The series converges absolutely.

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