I received a task to find out whether the following series converges:
$$\sum_{x=1}^\infty\sin(x)$$
On first look it seems simple, but as I keep thinking about it, there's not a single lemma or criterion that I can use to tackle the problem.
D'alembert ? Doesn't work: The following is meaningless IMHO: $\lim\limits_{x \to \infty}{}\frac{\sin(x+1)}{\sin(x)}$
That series isn't monotonic… you can't understand the rules for when will a member of the series be negative or positive.
All I know is that $\sin(x)$ is blocked between (-1) and 1.
Though it's easy to see that $ \sum\limits_{x=1}^\infty\lvert\sin(x) \rvert$ diverges.
May I use Leibniz formula for $\pi$ in order to construct 2 subseries:
- One that shows that $\sin(x)$ converges to the limit 1
- Another one that shows that $\sin(x)$ converges to the limit 0
And we know that a series can't converge to 2 different numbers, hence it diverges?
Best Answer
The sequence $(\sin n)$ doesn't converge to $0$ so the given series is divergent.