For the series

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

Problem

For the series ∑n=1∞sin⁡ ⁣(πn)\displaystyle \sum_{n=1}^{\infty} \sin\!\left(\dfrac{\pi}{n}\right), the limit of its terms is lim⁡n→∞sin⁡(π/n)=0\lim_{n\to\infty} \sin(\pi/n)=0. Explain, in one concise sentence, why this fact alone does not decide whether the series converges.

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