Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ converge?

Question

I am shockingly terrible at determining whether or not infinite series converge or not... I'm stuck on the problem:

Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ converge?

I attempted solving it using Taylor approximations:

$$\sin \frac{1}{k} \approx \frac{1}{k} - \frac{1}{3!k^3} + O(\frac{1}{k^5})$$ $$\arctan \frac{1}{k} \approx \frac{1}{k} - \frac{1}{3k^3} + O(\frac{1}{k^5})$$ $$\sin \frac{1}{k} - \arctan \frac{1}{k} \approx \frac{1}{k} - \frac{1}{3!k^3} + O(\frac{1}{k^5}) - \left(\frac{1}{k} - \frac{1}{3k^3} + O(\frac{1}{k^5}) \right) \approx \frac{1}{6k^3} + O(\frac{1}{k^5})$$

$$\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k}) \approx \sum_{k=1}^{\infty} (\frac{1}{6k^3} + O(\frac{1}{k^5})) < \infty$$ So the series converges. $\square$

My questions are:

1. Is my answer correct?

2. Is there anything wrong with using Taylor approximations to determine convergence?

3. Is there a rule of thumb for how to tackle questions which ask you to determine the convergence or divergence of series? If there is, please share - I could really use the help.

Answer 1

Your strategy does work. Why? Because the terms beyond $1/k^3$ are an arbitrarily small fraction of that term for sufficiently large $k$, so their contribution to large-$k$ terms are bounded on both sides by multiples of the $1/k^3$ part.