Chapter 7: Q. 20 (page 615)

Find two convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = L$ and $\sum_{k = 0}^{\infty} b_{k} = M$ with all positive terms such that $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}}$ diverges.

Short Answer

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Ans:

part (a). The convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = \sum_{k = 0}^{\infty} \frac{1}{2^{k}}$

part (b). The convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = \sum_{k = 0}^{\infty} \frac{1}{4^{k}}$

part (c). The series is diverges $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}} = \sum_{k = 0}^{\infty} 2^{k}$

Step by step solution

Step 1. Given information:

Consider the two convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = L$ and $\sum_{k = 0}^{\infty} b_{k} = M$ with all positive terms that localid="1649338303873" $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}}$ diverges.

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

Consider the geometric series $\sum_{k = 0}^{\infty} a_{k} = \sum_{k = 0}^{\infty} \frac{1}{2^{k}}$ .

The series $\sum_{k = 0}^{\infty} \frac{1}{2^{k}}$ 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, localid="1649339670370" $\sum_{k = 0}^{\infty} a_{k} = \sum_{k = 0}^{\infty} \frac{1}{2^{k}}$ is convergent.

Step 3. Finding the convergent geometric series ∑k=0∞bk

Consider the geometric series

\sum_{k = 0}^{\infty} b_{k} = \sum_{k = 0}^{\infty} \frac{1}{4^{k}}

The series

\sum_{k = 0}^{\infty} \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,

\sum_{k = 0}^{\infty} a_{k} = \sum_{k = 0}^{\infty} \frac{1}{4^{k}}

is convergent.

Step 4. Finding the converges geometric series of ∑k=0∞akbk with all positive terms:

$\sum_{k = 0}^{\infty} 2^{k}$ The series $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}}$ is

localid="1649340360408" $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}} = \sum_{k = 0}^{\infty} \frac{4^{k}}{2^{k}} = \sum_{k = 0}^{\infty} 2^{k}$

The series is $\sum_{k = 0}^{\infty} 2^{k}$ a geometric series with common ratio localid="1649340245008" $r = 2$ , which is less than 1 . The geometric series with ratio greater than 1 is diverges.

Therefore, $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}} = \sum_{k = 0}^{\infty} 2^{k}$ is diverges.

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Find two convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = L$ and $\sum_{k = 0}^{\infty} b_{k} = M$ with all positive terms such that $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}}$ diverges.

Short Answer

Step by step solution

Step 1. Given information:

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

Step 3. Finding the convergent geometric series ∑k=0∞bk

Step 4. Finding the converges geometric series of ∑k=0∞akbk with all positive terms:

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Find two convergent geometric series ∑k=0∞ak=Land ∑k=0∞bk=Mwith all positive terms such that ∑k=0∞akbkdiverges.

Short Answer

Step by step solution

Step 1. Given information:

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

Step 3. Finding the convergent geometric series ∑k=0∞bk

Step 4. Finding the converges geometric series of ∑k=0∞akbk with all positive terms:

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Most popular questions from this chapter

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Probability and Statistics

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Find two convergent geometric series $\sum_{k = 0}^{\infty} a_{k} = L$ and $\sum_{k = 0}^{\infty} b_{k} = M$ with all positive terms such that $\sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}}$ diverges.