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Essential Calculus: Early Transcendentals
Found in: Page 443
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

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

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Short Answer

Determine whether the geometric series is convergent or divergent..If it is convergent,find its sum.

\(\sum\limits_{n = 1}^\infty {\frac{{{{( - 3)}^{n - 1}}}}{{{4^n}}}} \)

The given series is Convergent and Sum=\(\frac{1}{7}\)..\(\)

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Finding the common ratio

Given Series \(\sum\limits_{n = 1}^\infty {\frac{{{{( - 3)}^{n - 1}}}}{{{4^n}}}} = \frac{1}{4} - \frac{3}{{16}} + \frac{9}{{64}} - \frac{{27}}{{256}} + ......\)

Common Ratio of this series=\(\frac{{{a_{n + 1}}}}{{{a_n}}}\)=\(\frac{{ - 3}}{{16}}*\frac{4}{1} = - \frac{3}{4}\)

Checking convergent

Since,\(|r| = \frac{3}{4} < 1\)

Therefore,the series is convergent..

Finding Sum..

Formula for infinite sum of a gp is =\(\frac{a}{{1 - r}}\)

Where a is first term and r is common ratio.

\({S_\infty } = \frac{{\frac{1}{4}}}{{1 + \frac{3}{4}}} = \frac{1}{7}\)

Hence,The Series is convergent and Sum=\(\frac{8}{3}\)

Most popular questions for Math Textbooks

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

\({a_n} = \frac{{{{\cos }^2}n}}{{{2^n}}}\)

Express the number as a ratio of integers. i) 10.135=10.135353535….

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?

\(\sum\limits_{n = 0}^\infty {{{( - 1)}^{n - 1}}n{e^{ - n}}{\rm{ (}}|{\rm{error}}|{\rm{ }} < 0.01)} \)

Find the values of x for which the series converges. Find the sum of the series for those values of x.

\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)

(a) Use a graph of \(y = \frac{1}{x}\)to show that if\({S_n}\)is the \({n^{th}}\)partial sum of the harmonic series, then\({S_n} \le 1 + \ln n\).

(b) The harmonic series diverges, but very slowly. Use part(a) to show that the sum of the first million term is less than\(15\)and the sum of the first billion terms is less than\(22\).
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