The series converges to

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The series ∑n=1∞(−1)n+112n−1\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{2n-1} converges to π/4\pi/4. Let S5S_5 and S7S_7 be the partial sums through the 5th and 7th terms, respectively. Calculate the alternating-series error bound for each partial sum and indicate which approximation is closer to π/4\pi/4.

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