Should I use the comparison test for the following series?

Question

Given the following series

$$\sum_{k=0}^\infty \frac{\sin 2k}{1+2^k}$$

I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would be to compare it to $\frac{1}{2^k}$ and since that is a geometric series that converges, my original series would converge as well. I'm not all too familiar with comparing series that have trig functions in them. Hope I'm going in the right direction

Thanks

Answer 1

You have the right idea, but you need to do a little more, since some of the terms are negative. Use your idea and the fact that $|\sin x|\le 1$ for all $x$ to show that

$$\sum_{k\ge 0}\frac{\sin 2k}{1+2^k}$$

is absolutely convergent, i.e., that

$$\sum_{k\ge 0}\left|\frac{\sin 2k}{1+2^k}\right|$$

converges.

Answer 2

Yes, you can compare with geometric series. Thus you will show that your serie converge absolutely.

Answer 3

Use asymptotic analysis: $$\Biggl\lvert\frac{\sin 2k}{1+2^k}\Biggr\rvert=_{\infty}O\Bigl(\frac1{2^k}\Bigr)$$ hence the series is absolutely convergent.

Answer 4

$\sum_{k\geq 0} \left| \frac{\sin(2k)}{1+2^k} \right| \leq \sum_{k\geq 0} \left| \frac{1}{2^k} \right| = 1$