Log In Start studying!

Select your language

Suggested languages for you:
Deutsch (DE)
Deutsch (UK)
Deutsch (US)

Americas

Europe

Answers without the blur. Sign up and see all textbooks for free! → Illustration

Q28E

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

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later
Illustration

Short Answer

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.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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.\(\)

Most popular questions for Math Textbooks

Determine whether the sequence converges or diverges. If it converges, find the limit.

\({a_n} = \frac{{{{( - 1)}^n}}}{{2\sqrt n }}\)

Find the Value of \(x\) for which the series converges. Find the sum of

the series for those values of \(x\)

\(\sum\limits_{n = 0}^\infty {{{( - 4)}^n}} {(x - 5)^n}\)

Let S be the sum of a series of \(\sum {{a_n}} \) that has shown to be convergent by the Integral Test and let f(x) be the function in that test. The remainder after n terms is

\({R_n} = S - {S_n} = {a_{n + 1}} + {a_{n + 2}} + {a_{n + 3}} + .......\)

Thus, Rn is the error made when Sn , the sum of the first n terms is used as an approximation to the total sum S.

(a) By comparing areas in a diagram like figures 3 and 4 (but with x ≥ n), show that

\(\int\limits_{n + 1}^\infty {f(x)dx \le {R_n} \le \int\limits_n^\infty {f(x)dx} } \)

(b) Deduce from part (a) that

\({S_n} + \int\limits_{n + 1}^\infty {f(x)dx \le S \le {S_n} + \int\limits_n^\infty {f(x)dx} } \)

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

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

\(\sum\limits_{n = 1}^\infty {\frac{{{{\left( {{\rm{ - }}1} \right)}^n}}}{{n{5^n}}}} \) \(\left( {\left| {error} \right| < 0.0001} \right)\)
Icon

Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

Decision Maths

Read explanation

Calculus

Read explanation

Probability and Statistics

Read explanation

Statistics

Read explanation

Mechanics Maths

Read explanation

Geometry

Read explanation

Pure Maths

Read explanation

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.