AP Calculus BC · Parametric, polar, and vector-valued functions

Using standard trigonometric limits and the limit comparison test, determine whether…

Problem

Using standard trigonometric limits and the limit comparison test, determine whether each series converges or diverges. (a) $\displaystyle\sum_{n=1}^{\infty}\sin\frac1n$. (b) $\displaystyle\sum_{n=1}^{\infty}\sin\frac1{n^2}$. (c) $\displaystyle\sum_{n=1}^{\infty}\Bigl(1-\cos\frac1n\Bigr)$. In each part, identify the comparison series and the value of the limiting ratio. Then briefly explain why the three conclusions are consistent with the small-angle relations $\sin\theta\sim\theta$ and $1-\cos\theta\sim\theta^2/2$.

Hint

Replace each trigonometric expression by its standard small-angle equivalent, then limit-compare with the matching $p$-series.

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