The alternating harmonic series is used to approximate

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

Problem

The alternating harmonic series ∑k=1∞(−1)k+11k\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1}\dfrac{1}{k} is used to approximate ln⁡2\ln 2. A student stops after the first three terms and obtains the partial sum S3S_{3}. Determine whether this approximation is guaranteed to be within 0.100.10 of ln⁡2\ln 2. (a) No; the maximum possible error is 0.200.20, which exceeds 0.100.10. (b) No; the maximum possible error is 0.250.25, which exceeds 0.100.10. (c) No; the maximum possible error is 0.1250.125, which exceeds 0.100.10. (d) Yes; the maximum possible error is 0.0830.083, which is less than 0.100.10.

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