Consider the series

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

Problem

Consider the series ∑k=1∞(−1)k+11k2\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1}\dfrac{1}{k^{2}}. Let S4S_{4} denote the sum of the first four terms of the series. Is S4S_{4} guaranteed to approximate the exact sum of the series within 0.050.05? (a) No, because the maximum possible error is 0.0625>0.050.0625>0.05. (b) Yes, because the maximum possible error is 0.040.04. (c) Yes, because the maximum possible error is 0.020.02. (d) No, because the maximum possible error is approximately 0.068>0.050.068>0.05.

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