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.

Short Answer

Find the values of x for which the series converges. Find the sum of the series for those values of x.
\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)

The given series converges for \( - 1 < x < 5\). Sum of series for these values of \(x = \frac{3}{{5 - x}}\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1

Finding the interval where the series converges by using the concept that an infinite geometric whose \(|r| < 1\) is convergent.

This series converges if we assume it to be of the form of an infinite geometric series, \(\sum\limits_{k = 0}^\infty {a{r^k} = \frac{a}{{1 - r}},|r| < 1} \)

Comparing the given series with the infinite geometric series, we observe that \(a = 1\) and \(r = \frac{{x - 2}}{3}\).

Putting |r|<1, \(|\frac{{x - 2}}{3}| < 1\)

ie \( - 1 < \frac{{x - 2}}{3} < 1\) \( \Rightarrow - 3 < x - 2 < 3 \Rightarrow - 1 < x < 5.\)

Therefore, the series converges when \(x\) belongs to \(( - 1,5).\)

Step2

Sum of an infinite geometric progression is given by \(\frac{a}{{1 - r}}\).

Finding sum of series when \(x\) is in the range \(( - 1,5).\)

So, we obtain:\(\frac{1}{{1 - (\frac{{x - 2}}{3})}} = \frac{3}{{5 - x}}\).

Find Study Materials for

Create Study Materials

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Find the values of x for which the series converges. Find the sum of the series for those values of x.
\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)

Step by Step Solution

Step 1

Step2

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Find the values of x for which the series converges. Find the sum of the series for those values of x.\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)

Step by Step Solution

Step 1

Step2

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Find the values of x for which the series converges. Find the sum of the series for those values of x.
\(\sum\nolimits_{n = 0}^\infty {\frac{{{{(x - 2)}^n}}}{{{3^n}}}} \)