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Q28E

Expert-verifiedFound in: Page 453

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.

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

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

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

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.

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.\(\)

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