Chapter 7: Q. 16 (page 656)

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} \frac{5^{k + 3}}{2^{3 k + 1}}$

Short Answer

Expert verified

The sum of the series is $\frac{500}{3}$ .

Step by step solution

Step 1. Given Information

The given series is $\sum_{k = 0}^{\infty} \frac{5^{k + 3}}{2^{3 k + 1}}$ .

Step 2. Determine if a geometric series

Find the ratio of the consecutive terms.

$\begin{array}{rcl} \frac{\frac{5^{k + 3 + 1}}{2^{3 (k + 1) + 1}}}{\frac{5^{k + 3}}{2^{3 (k) + 1}}} & = & \frac{5^{k + 3 + 1 - k - 3}}{2^{3 (k + 1) + 1 - 3 k - 1}} \\ = & \frac{5}{2^{3}} \\ = & \frac{5}{8} \end{array}$

Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratio $r = \begin{array}{rcl} \frac{5}{8} \end{array}$ .

Step 3. Determine the convergence

Determine the first term and the common ratio.

$\begin{array}{rcl} c & = & \frac{5^{0 + 3}}{2^{3 (0) + 1}} \\ = & \frac{5^{3}}{2} \\ = & \frac{125}{2} \\ r & = & \frac{5}{8} \end{array}$

If $|r| < 1$ , the series converges to $\frac{c}{1 - r}$ .

$\begin{array}{rcl} \frac{c}{1 - r} & = & \frac{\frac{125}{2}}{1 - \frac{5}{8}} \\ = & \frac{\frac{125}{2}}{\frac{3}{8}} \\ = & \frac{500}{3} \end{array}$

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} \frac{5^{k + 3}}{2^{3 k + 1}}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges. ∑k=0∞5k+323k+1

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 0}^{\infty} \frac{5^{k + 3}}{2^{3 k + 1}}$