Chapter 12: Q 63. (page 825)

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1}$

Short Answer

Expert verified

The given infinite geometric series $\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1}$ is convergent and its sum is $\frac{18}{5}$ .

Step by step solution

Step 1. Write the given information.

The given geometric series is:

$\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1}$

Determine the first term and common ratio.

The first term is $(a_{1} = 6)$

The common ratio is the ratio of successive terms:

$r = \frac{- 2}{3}$

Step 3. Determine whether the infinite geometric series converges or diverges.

If $|r| < 1$ then the infinite geometric series $\sum_{k = 1}^{\infty} a_{1} r^{k - 1}$ converges.

As we can see that $|- \frac{2}{3}| = \frac{2}{3} and \frac{2}{3} < 1$ , therefore the given series is convergent.

Step 4. Find the sum of the series.

Use the formula to find the sum of the given geometric series $\sum_{k = 1}^{\infty} a_{1} r^{k - 1} = \frac{a_{1}}{1 - r}$ ,

$\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1} = \frac{6}{(1 - (\frac{- 2}{3}))} = \frac{6}{\frac{5}{3}} = \frac{18}{5}$

Therefore, the sum of the infinite geometric series is $\frac{18}{5}$ .

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In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1}$

Short Answer

Step by step solution

Step 1. Write the given information.

Determine the first term and common ratio.

Step 3. Determine whether the infinite geometric series converges or diverges.

Step 4. Find the sum of the series.

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Most popular questions from this chapter

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In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.∑k=1∞6-23k-1

Short Answer

Step by step solution

Step 1. Write the given information.

Determine the first term and common ratio.

Step 3. Determine whether the infinite geometric series converges or diverges.

Step 4. Find the sum of the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Geometry

Statistics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} 6 {(- \frac{2}{3})}^{k - 1}$