Question

Determine whether the series is convergent or divergent:\(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \).

Short answer

The series \(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \) is convergent.

Step-by-step solution

Comparing the series:

In this series, \(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \)

Since,\(\sin n \le 1\)

So, \(\frac{{1 + \sin n}}{{{{10}^n}}} < \frac{2}{{{{10}^n}}}\)

Convergence of geometric series:

Now, the series \(\sum\limits_{n = 0}^\infty {\frac{2}{{{{10}^n}}}} \) is a geometric series with \(n = \frac{1}{{10}}\).

Since \(|n| < 1\)

So, \(\sum\limits_{n = 0}^\infty {\frac{2}{{{{10}^n}}}} \) series is convergent.

Comparison tests:

As \(\sum\limits_{n = 0}^\infty {\frac{2}{{{{10}^n}}}} \) series is convergent, therefore,

By comparison test, the series

\(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \) is also convergent.

Hence, \(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \)is also convergent.\(\)