Convergence/divergence series

Question

Determine whether the following is converging or diverging

$$\sum_{i=1}^∞ \frac{\sin(1/i)}{\sqrt{i}}$$

I tried the following, but not sure whether it's correct:

$$\frac{\sin(1/i)}{\sqrt i}$$ <= $$\frac {1}{n\sqrt n}$$

Hence by the nth term test, $$\frac {1}{n\sqrt n}$$ is convergent, hence the series converges

But I am not sure how to prove $$\frac{\sin(1/i)}{\sqrt i}$$ is less than or = to $$\frac {1}{n\sqrt n}$$

Answer 1

Simply note that

$$\frac{\sin(1/i)}{\sqrt{i}}\sim \frac{1}{i\sqrt{i}}$$

then the given series converges by limit comparison test with $\sum \frac{1}{i\sqrt{i}}$.

Answer 2

Hint. Every term of your sum is non-negative. Use the inequality $\sin x\le x$ for $x\ge 0$.