Q. 18 Find two convergent geometric se... [FREE SOLUTION]

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Chapter 7: Q. 18 (page 615)

Find two convergent geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = L$ and ${鈭�}_{k = 0}^{鈭�} b_{k} = M$ with all positive terms such that ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}}$ bk converges but whose sum is not $\frac{L}{M}$ .
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Short Answer

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Ans:

part (a). The convergent geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}}$

part (b). The convergent geometric series ${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$

part (c). The serise is converge ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$

part (d). The series ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ converge to the sum 2

The series ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ do not converge to the sum of $\frac{{鈭�}_{k = 0}^{鈭�} a_{k}}{{鈭�}_{k = 0}^{鈭�} b_{k}}$ .

Step by step solution

Step 1. Given information:

Consider the two convergent geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = L$ and ${鈭�}_{k = 0}^{鈭�} b_{k} = M$ such that ${鈭�}_{k = 0}^{鈭�} a_{k} 路 b_{k}$ converge.

Step 2. Finding the convergent geometric series ∑k=0∞ak=L

Consider the geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}}$ .

The series ${鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}}$ is a geometric series with common ratio $r = \frac{1}{4}$ , which is less than 1 .

The geometric series with ratio less than 1 is convergent.

Therefore, ${鈭�}_{k = 0}^{鈭�} a_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}}$ is convergent.

Step 3. Checking the convergent geometric series :

$s = \frac{1}{1 - \frac{1}{4}} (S u m o f g e o m e t r i c s e r i e s) = \frac{4}{4 - 1} = \frac{4}{3} 鈭� t h e s e r i e s {鈭�}_{k = 0}^{鈭�} a_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}} i s c o n v e r g e t o t h e s u m \frac{4}{3}$

Step 4. Finding the convergent geometric series ∑k=0 ∞bk =M

Consider the geometric series ${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ .

The series ${鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ is a geometric series with common ratio $r = \frac{1}{2}$ , which is less than 1 . The geometric series with ratio less than 1 is convergent.

Therefore, ${鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ is convergent.

Step 6. Checking the convergent geometric series :

$s = \frac{1}{1 - \frac{1}{2}} (S u m o f g e o m e t r i c s e r i e s) = \frac{2}{2 - 1} = \frac{2}{1} 鈭� t h e s e r i e s {鈭�}_{k = 0}^{鈭�} b_{k} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}} i s c o n v e r g e t o t h e s u m 2$

Step 6. Finding series converge to LM :

The series ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}}$ is

${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{2^{k}}{4^{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$

The series ${鈭�}_{k = 0}^{鈭�} \frac{1}{4^{k}}$ is a geometric series with common ratio $r = \frac{1}{2}$ , which is less than 1 . The geometric series with ratio less than 1 is convergent.

Therefore, ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}}$ is convergent.

Step 7. Checking the convergent geometric series :

$s = \frac{1}{1 - \frac{1}{2}} (S u m o f g e o m e t r i c s e r i e s) = \frac{2}{2 - 1} = \frac{2}{1} 鈭� t h e s e r i e s {鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}} = {鈭�}_{k = 0}^{鈭�} \frac{1}{2^{k}} i s c o n v e r g e t o t h e s u m 2$

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Find two convergent geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = L$ and ${鈭�}_{k = 0}^{鈭�} b_{k} = M$ with all positive terms such that ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}}$ bk converges but whose sum is not $\frac{L}{M}$ .
.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Finding the convergent geometric series ∑k=0∞ak=L

Step 3. Checking the convergent geometric series :

Step 4. Finding the convergent geometric series ∑k=0 ∞bk =M

Step 6. Checking the convergent geometric series :

Step 6. Finding series converge to LM :

Step 7. Checking the convergent geometric series :

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Find two convergent geometric series鈭�k=0鈭�ak=Land 鈭�k=0鈭�bk=Mwith all positive terms such that 鈭�k=0鈭�akbkbk converges but whose sum is not LM..

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Finding the convergent geometric series ∑k=0∞ak=L

Step 3. Checking the convergent geometric series :

Step 4. Finding the convergent geometric series ∑k=0 ∞bk =M

Step 6. Checking the convergent geometric series :

Step 6. Finding series converge to LM :

Step 7. Checking the convergent geometric series :

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Calculus

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find two convergent geometric series ${鈭�}_{k = 0}^{鈭�} a_{k} = L$ and ${鈭�}_{k = 0}^{鈭�} b_{k} = M$ with all positive terms such that ${鈭�}_{k = 0}^{鈭�} \frac{a_{k}}{b_{k}}$ bk converges but whose sum is not $\frac{L}{M}$ .
.