Answer to Question #264624 in Real Analysis for Sarita bartwal

Question

∞Σn=1 sin(1/n) is a convergent series.

True or false with full explanation

Answer 1

It is known that the harmonic series $\sum\limits_{n = 1}^\infty {\frac{1}{n}}$ diverges

Since

$\mathop {\lim }\limits_{n \to \infty } \frac{{\sin \frac{1}{n}}}{{\frac{1}{n}}} = \mathop {\lim }\limits_{\frac{1}{n} \to 0} \frac{{\sin \frac{1}{n}}}{{\frac{1}{n}}} = 1$ and $0 < 1 < \infty$ than, by Limit comparison test, the series $\sum\limits_{n = 1}^\infty {\sin \frac{1}{n}}$ also diverges.

Answer: False