Define the series (S= _ k=1 ^ (-1)^ k+1 k^ 2 )

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define the series S=∑k=1∞(−1)k+1k2S=\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{2}}. A numerical analyst wants an approximation whose maximum possible error does not exceed 0.0005. (a) Determine the least positive integer NN such that the partial sum SN=∑k=1N(−1)k+1k2S_N=\displaystyle\sum_{k=1}^{N}\frac{(-1)^{k+1}}{k^{2}} is guaranteed to differ from SS by less than 0.0005. (b) Compute SNS_N and SN+1S_{N+1} to six decimal places. (c) Using only information available from the series itself and without evaluating additional terms, identify which of the two values obtained in part (b) is guaranteed to be closer to SS. Provide a clear argument.

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