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Expert-verified Found in: Page 453 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # Determine whether the series is convergent or divergent:$$\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}}$$.

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

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## 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. ### Want to see more solutions like these? 