A student asserts that the alternating series error bound may be used to estimate the…
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A student asserts that the alternating series error bound may be used to estimate the remainder of n=1∑∞(−1)nnsin(1/n) because the limit of the terms is zero. Which of the following comments most accurately evaluates the student’s assertion?
(a) The claim is valid because (−1)n ensures alternation and the limit of the terms is zero, so the next term always bounds the remainder.
(b) The claim is invalid; a decreasing sequence of term magnitudes is also required, and this has not been established.
(c) The claim fails because sin(1/n) is always positive, meaning the series is not truly alternating.
(d) The claim is unnecessary; since n=1∑∞nsin(1/n) converges absolutely, every remainder is automatically smaller than the next term.