Q. 13 - Calculus

Question

Find a series $\sum _{k=1}^{\mathrm{\infty }} a_k$ with all non - zero terms that converges to 1 ,

Step-by-Step Solution

Answer

Series converges to 1 .

Step 1. Given information

We have been given a series$\sum _{k=1}^{\mathrm{\infty }} a_k$ and to find out the convergence of series .

Step 2. Checking whether the series is convergent .

Consider the series $\sum _{k=1}^{\mathrm{\infty }} a_k$ 

The objective is to find the geometric series $\sum _{k=1}^{\mathrm{\infty }} a_k$ with all non zero terms

Consider the series $\sum _{k=1}^{\mathrm{\infty }} a_k=\sum _{k=1}^{\mathrm{\infty }} \frac{1}{2^k}$

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 ,$\sum _{k=1}^{\mathrm{\infty }} a_k=\sum _{k=1}^{\mathrm{\infty }} \frac{1}{2^k}$ is convergent .

Step 3. Value of convergence

The series converges $\sum _{k=1}^{\mathrm{\infty }} a_k=\sum _{k=1}^{\mathrm{\infty }} \frac{1}{2^k}$ to

$\begin{array}{r}S=\frac{\frac{1}{2}}{1-\frac{1}{2}}\\ =\frac{{\displaystyle \frac{1}{2}}}{\frac{1}{2}}\end{array}$

           =$1$

Hence the series converges to $1$