Chapter 7: Q. 13 (page 614)

Find a series $\sum_{k = 1}^{\infty} a_{k}$ with all non - zero terms that converges to 1 ,

Short Answer

Expert verified

Series converges to 1 .

Step by step solution

Step 1. Given information

We have been given a series $\sum_{k = 1}^{\infty} a_{k}$ and to find out the convergence of series .

Step 2. Checking whether the series is convergent .

Consider the series $\sum_{k = 1}^{\infty} a_{k}$

The objective is to find the geometric series $\sum_{k = 1}^{\infty} a_{k}$ with all non zero terms

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

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 , $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{2^{k}}$ is convergent .

Step 3. Value of convergence

The series converges $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{2^{k}}$ to

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

= $1$

Hence the series converges to $1$

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Find a series $\sum_{k = 1}^{\infty} a_{k}$ with all non - zero terms that converges to 1 ,

Short Answer

Step by step solution

Step 1. Given information

Step 2. Checking whether the series is convergent .

Step 3. Value of convergence

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Short Answer

Step by step solution

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Step 2. Checking whether the series is convergent .

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Find a series $\sum_{k = 1}^{\infty} a_{k}$ with all non - zero terms that converges to 1 ,