Which of the following best explains why the series converges to a finite value?
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Which of the following best explains why the series 6⋅10−2+6⋅10−4+6⋅10−6+… converges to a finite value?
(a) Because 6⋅10−2n=102n6 is of the form npk with p=2>1, the series is a p-series that converges.
(b) Because the terms decrease to 0, adding them infinitely many times must give a finite sum, so the series converges.
(c) The absolute value of its common ratio is 10−2<1, so an infinite geometric series formula applies and the sum converges.
(d) The ratio test gives n→∞limanan+1=10−2<1. Since the limit is less than 1, the ratio test is inconclusive, but the terms go to 0, so the series converges.