Trying to see if $$\sum\limits_{n=1}^{\infty}\tan(1/n)$$ converges or diverges. As $n \to \infty$, $\tan(1/n) \to 0$, so inconclusive. Ratio test was inconclusive, root test doesn't look good for this one, and it's clearly not an alternating series, so that leads to perhaps some comparison arguments. Not sure what to use for comparison, other than maybe $\sin(1/n)$, but I know nothing about that series.
[Math] Convergence/Divergence of the series $\sum\limits_{n=1}^{\infty}\tan(1/n)$
calculussequences-and-series
Best Answer
Note that $$ \lim_{n\to\infty}\frac{\tan(1/n)}{1/n} = \lim_{t\to0}\frac{\tan(t)}{t} \stackrel{L}{=} \lim_{t\to0}\frac{\sec^2(t)}{1} = 1 $$ The limit comparision test then implies that either both $\sum1/n$ and $\sum\tan(1/n)$ converge or both diverge. But $\sum 1/n$ diverges so $\sum \tan(1/n)$ diverges.