Chapter 7: Q. 19 (page 652)

Give an example of divergent series $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ such that the series role="math" localid="1649434191353" $\sum_{k = 1}^{\infty} a_{k} b_{k}$ converges.

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} a_{k} b_{k} = \sum_{k = 1}^{\infty} (\frac{1}{k^{2}})$ is convergent.

Step by step solution

Step 1. Given information

$\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ are divergent.

Step 2. Check if ∑akk=1∞ converges

Calculating $\frac{a_{k + 1}}{a_{k}}$ ,

a_{k} = \frac{1}{k}

$a_{k + 1} = \frac{1}{(k + 1)}$

$\begin{aligned} \frac{a_{k + 1}}{a_{k}} = \frac{\frac{1}{(k + 1)}}{\frac{1}{k}} \\ = \frac{k}{(k + 1)} \end{aligned}$

Now, taking limits,

$\begin{aligned} lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = lim_{k \to \infty} \frac{k}{(k + 1)} \\ = (lim_{k \to \infty} \frac{k}{(k + 1)}) \\ = \infty \end{aligned}$

Hence, the series role="math" localid="1649434828493" ${\sum a_{k}}_{k = 1}^{\infty} = \sum_{k = 1}^{\infty} \frac{1}{k}$ diverges.

Similarly, ${\sum b_{k}}_{k = 1}^{\infty} = \sum_{k = 1}^{\infty} \frac{1}{k}$ isdivergent.

Step 3. Check if ∑akbkk=1∞ is divergent or convergent

$\begin{aligned} \sum_{k = 1}^{\infty} a_{k} b_{k} = \sum_{k = 1}^{\infty} (\frac{1}{k}) (\frac{1}{k}) \\ = \sum_{k = 1}^{\infty} (\frac{1}{k^{2}}) \end{aligned}$

Hence, $\sum_{k = 1}^{\infty} a_{k} b_{k}$ is convergent.

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Give an example of divergent series $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ such that the series role="math" localid="1649434191353" $\sum_{k = 1}^{\infty} a_{k} b_{k}$ converges.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check if ∑akk=1∞ converges

Step 3. Check if ∑akbkk=1∞ is divergent or convergent

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Give an example of divergent series ∑k=1∞akand ∑k=1∞bksuch that the series role="math" localid="1649434191353" ∑k=1∞akbkconverges.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check if ∑akk=1∞ converges

Step 3. Check if ∑akbkk=1∞ is divergent or convergent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Give an example of divergent series $\sum_{k = 1}^{\infty} a_{k}$ and $\sum_{k = 1}^{\infty} b_{k}$ such that the series role="math" localid="1649434191353" $\sum_{k = 1}^{\infty} a_{k} b_{k}$ converges.