Chapter 7: Q 58. (page 615)

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges to $\frac{30}{11}$ .

Step by step solution

Step 1. Given information.

Given a series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ .

Step 2. Find if the series converges or not.

The series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ can be expressed as ${鈭� 5}_{k = 0}^{鈭�} {(\frac{- 5}{6})}^{k}$ .

The series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ is in the standard form ${鈭�}_{k = 0}^{鈭�} c r^{k}$ for a geometric series with $c = 5$ and $r = - \frac{5}{6}$ .

The geometric series converges if and only if $|r| < 1$ .

Since localid="1648983652785" $r = - \frac{5}{6}$ , it follows that the series converges.

Step 3. Find the value to which the series converges.

If the geometric series ${鈭�}_{k = 0}^{鈭�} c r^{k}$ converges, it converges to $\frac{c}{1 - r}$ .

So, the series localid="1648983661968" ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges to localid="1648983659158" $\frac{5}{1 - (- \frac{5}{6})}$ , that is localid="1648983666395" $\frac{30}{11}$ .

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91影视

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

Step 3. Find the value to which the series converges.

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91影视

Determine whether the series鈭�k=0鈭�5k+1-6k converges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

Step 3. Find the value to which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Decision Maths

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges or diverges. Give the sum of the convergent series.