A series of positive terms satisfies

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

Problem

A series of positive terms satisfies lim⁡n→∞an+1an=0.8\displaystyle \lim_{n\to\infty}\dfrac{a_{n+1}}{a_n}=0.8. Which of the following conclusions about ∑n=1∞an\sum_{n=1}^{\infty} a_n is most accurate? (a) The series converges conditionally but not absolutely because the ratio limit is less than 1.\text{The series converges conditionally but not absolutely because the ratio limit is less than }1. (b) The series converges absolutely because the ratio limit is less than 1.\text{The series converges absolutely because the ratio limit is less than }1. (c) The series diverges because its terms do not approach 0 fast enough (each term is about 80% of the previous).\text{The series diverges because its terms do not approach }0\text{ fast enough (each term is about }80\%\text{ of the previous).} (d) The ratio test is inconclusive since the limit is not equal to 1.\text{The ratio test is inconclusive since the limit is not equal to }1.

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