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Chapter 7: Q. 13 (page 614)

Find a series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ with all non - zero terms that converges to 1 ,

Short Answer

Expert verified

Series converges to 1 .

Step by step solution

01

Step 1. Given information

We have been given a series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ and to find out the convergence of series .

02

Step 2. Checking whether the series is convergent .

Consider the series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$

The objective is to find the geometric series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ with all non zero terms

Consider the series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k} = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{2^{k}}$

The series is a geometric series with geometric ratio $\frac{1}{2}$ which is less than $1$ .

The series with geometric ratio less than 1 is convergent in nature .

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

03

Step 3. Value of convergence

The series converges ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k} = {鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{2^{k}}$ to

role="math" localid="1649674645349" $\begin{aligned} S = \frac{\frac{1}{2}}{1 鈭� \frac{1}{2}} \\ = \frac{\frac{1}{2}}{\frac{1}{2}} \end{aligned}$

= $1$

Hence the series converges to $1$

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Most popular questions from this chapter

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

Explain why, if n is an integer greater than 1, the series ${鈭�}_{k = 1}^{鈭�} \frac{1}{\sqrt[n]{k}}$ diverges.

Determine whether the series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ converges or diverges. Give the sum of the convergent series.

Let $a : [1, 鈭�) 鈫� 鈩�$ be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral ${鈭�}_{1}^{鈭�} a (x) d x$ converges, then the series localid="1649180069308" ${鈭�}_{k = 1}^{鈭�} a (k)$ does too.

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k^{2} + 1}{k!}$

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