Series Convergence/Divergence Proofs

[chimychang]

$$\sum_{n=1}^{\infty} n \sin(\frac{1}{n})$$

I rewrote the sum as $$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\frac{1}{n}}$$

Then I applied the Nth term test and used L'Hoptials rule so $$\lim_{n\to\infty} \frac{\cos(\frac{1}{n})\frac{-1}{n^2}}{\frac{-1}{n^2}}$$

The $$\frac{-1}{n^2}$$ cancel out and the $$lim_{n\to\infty} \cos(\frac{1}{n})$$ is 1 which by the nth term test is divergent. Is that a legitimate proof of divergence?

 

[Char. Limit]

Yes.