Question

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

Short answer

Series converges to 1 .

Step-by-step solution

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

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

=$1$

Hence the series converges to$1$