# Is the following series convergent?

## Answer True/False to the statements. I found that the limit is 0, and have tried ratio test (L=1) and comparison test, but am unsure how to prove it converges/diverges.

Jun 13, 2018

The series:

${\sum}_{n \to \infty} \frac{1}{n + \sin n}$

is divergent.

#### Explanation:

As:

$- 1 \le \sin x \le 1$

we have that:

$\frac{1}{n + 1} \le \frac{1}{n + \sin n} \le \frac{1}{n - 1}$

Based on the squeeze theorem we can therefore immediately determine that:

${\lim}_{n \to \infty} \frac{1}{n + \sin n} = 0$

Consider now the series:

${\sum}_{n \to \infty} \frac{1}{n + 1}$

As:

${\lim}_{n \to \infty} \frac{\frac{1}{n}}{\frac{1}{n + 1}} = {\lim}_{n \to \infty} \frac{n + 1}{n} = 1$

based in the limit comparison test it must have the same character as the harmonic series that we know to be divergent.

But as:

$\frac{1}{n + 1} \le \frac{1}{n + \sin n}$

then also the series:

${\sum}_{n \to \infty} \frac{1}{n + \sin n}$

is divergent by direct comparison.