Q. 89 Let鈭慿=0鈭瀋rkand聽鈭慿=0鈭瀊vk... [FREE SOLUTION]

Chapter 7: Q. 89 (page 616)

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

Short Answer

Expert verified

${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}} = \frac{c}{b} {鈭�}_{k = 0}^{鈭�} {(\frac{r}{v})}^{k}$ and so the series is in geometric progression.

The condition for the series ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ to be convergent is $|\frac{r}{v}| < 1$ .

Step by step solution

Step 1. Given Information.

${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent series.

Step 2. Prove that the series is geometric.

The expanded form of the series ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ will be:

${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}} = \frac{c}{b} (1 + \frac{r}{v} + {(\frac{r}{v})}^{2} + . . .) = \frac{c}{b} {鈭�}_{k = 0}^{鈭�} \frac{r^{k}}{v^{k}} = \frac{c}{b} {鈭�}_{k = 0}^{鈭�} {(\frac{r}{v})}^{k}$

Therefore the series is in geometric progression.

Step 3. Conditions met for the series to converge.

For any geometric series to converge, its common ratio must be less than 1.

The series ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is in geometric progression and its common ratio is $\frac{r}{v}$ .

So the condition for the series to converge is given as $|\frac{r}{v}| < 1$ .

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Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove that the series is geometric.

Step 3. Conditions met for the series to converge.

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91影视

Let鈭�k=0鈭�crkand鈭�k=0鈭�bvk be two convergent geometric series. If b and v are both nonzero, prove that 鈭�k=0鈭�crkbvk is a geometric series. What condition(s) must be met for this series to converge?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Prove that the series is geometric.

Step 3. Conditions met for the series to converge.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?