For the alternating series

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

Problem

For the alternating series ∑k=1∞(−1)k+11k2\displaystyle \sum_{k=1}^{\infty} (-1)^{k+1}\frac{1}{k^{2}}, let SS denote the exact sum and SnS_{n} the sum of its first nn terms. Explain in one sentence why ∣S−Sn∣|S-S_{n}| cannot exceed the magnitude of the (n+1)(n+1)st term.

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