A student claims that for any convergent alternating series with positive terms…
MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A student claims that for any convergent alternating series with positive terms decreasing to 0, adding one more term to a partial sum guarantees a smaller maximum possible error than before. Which of the following best justifies the student’s claim?
(a) Because the Ratio Test guarantees ∣an+1∣<∣an∣ and convergence, each new term must reduce the remainder by roughly that same ratio.
(b) For an alternating series whose terms decrease in magnitude, the Alternating Series Remainder Theorem states that ∣S−Sn∣≤∣an+1∣, and since ∣an+1∣<∣an∣, adding the (n+1)st term lowers the bound.
(c) The partial sums of any alternating series form a monotone sequence, so every additional term moves the sum closer to the limit and reduces the error.
(d) All alternating series are geometric, so each successive term is a constant fraction of the preceding term, making every new term a tighter bound on the error.