Chapter 8.1, Problem 108E

To determine

To write: the sum under sigma notation and evaluate it by graphing utility.

Answer to Problem 108E

The sum can be written as ${\displaystyle \sum _{j=0}^7{\left(-1\right)}^j{\left(\frac{1}{2}\right)}^j}$ .

And

  ${\displaystyle \sum _{j=0}^7{\left(-1\right)}^j{\left(\frac{1}{2}\right)}^j}=\frac{85}{128}$

Explanation of Solution

Given information:

A finite series expression is given as

  $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\mathrm{...}-\frac{1}{128}$

Concept used:

A finite series of the form $a_1+a_2+a_3+\mathrm{...}+a_n$ can be written in sigma notation as

  ${\displaystyle \sum _{j=1}^n{a}_j}$

Calculation:

Consider the series expression.

  $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\mathrm{...}-\frac{1}{128}$

Here, the series can be rewritten as

  ${\left(-1\right)}^0{\left(\frac{1}{2}\right)}^0+{\left(-1\right)}^1{\left(\frac{1}{2}\right)}^1+{\left(-1\right)}^2{\left(\frac{1}{2}\right)}^2+{\left(-1\right)}^3{\left(\frac{1}{2}\right)}^3+\mathrm{...}+{\left(-1\right)}^7{\left(\frac{1}{2}\right)}^7$

So, the lower index is 0 and upper index is 7.

Consider the variable $j$ to write the index lower limit and upper limit for the sum.

So, the series can be written in sigma notation as

  ${\left(-1\right)}^0{\left(\frac{1}{2}\right)}^0+{\left(-1\right)}^1{\left(\frac{1}{2}\right)}^1+{\left(-1\right)}^2{\left(\frac{1}{2}\right)}^2+{\left(-1\right)}^3{\left(\frac{1}{2}\right)}^3+\mathrm{...}+{\left(-1\right)}^7{\left(\frac{1}{2}\right)}^7={\displaystyle \sum _{j=0}^7{\left(-1\right)}^j{\left(\frac{1}{2}\right)}^j}$

Follow the below steps to evaluate the sum by graphing utility

1. Choose the SUM feature from MATH menu
2. Choose SEQUENCE feature from OPERATION menu
3. Enter the expression for the sequence ${\left(-1\right)}^j{\left(\frac{1}{2}\right)}^j$ , variable of the sequence $\left(j\right)$ , lower limit of the summation $\left(0\right)$ , upper limit of the summation $\left(7\right)$
4. Press the button ENTER

The calculator will display the result as $\frac{85}{128}$ .

Hence, the value of the sum is

  ${\displaystyle \sum _{j=1}^7{\left(-1\right)}^j{\left(\frac{1}{2}\right)}^j}=\frac{85}{128}$