Show that the series diverges

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

Problem

Show that the series ∑n=1∞sin⁡ ⁣(1n)\displaystyle\sum_{n=1}^{\infty}\sin\!\left(\frac{1}{n}\right) diverges. Provide inequalities giving both an upper and a lower bound for sin⁡ ⁣(1n)\sin\!\left(\frac{1}{n}\right) that enable a direct comparison with a pp-series, then state your conclusion.

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