Chapter 9.1, Problem 83E

To determine

To represent:The sum of the series $3-9+27-81+243-729$ in the form ofsigma notation.

Answer to Problem 83E

The sum of the series $3-9+27-81+243-729$ is ${\displaystyle \sum _{i=1}^6{\left(-1\right)}^{i+1}{3}^i}$ .

Explanation of Solution

Given information:

The given series is $3-9+27-81+243-729$ .

Calculation:

Let’s a sequence has $n$ terms and it can be represented as follows.

  ${\displaystyle \sum _{i=1}^n{a}_i}=a_1+a_2+a_3+a_4+\mathrm{...}+a_n$

Here, the index of summation is $i$ , the lower limit of summation is 1 and the upper limit of summation is $n$ .

In the given sequence, there are 6 terms. So, the sum of the series is,

  $\begin{array}{c}3-9+27-81+243-729=3-3^2+3^3-3^4+3^5-3^6\\ ={\left(-1\right)}^2{3}^1+{\left(-1\right)}^3{3}^2+{\left(-1\right)}^4{3}^3+{\left(-1\right)}^5{3}^4+{\left(-1\right)}^6{3}^5+{\left(-1\right)}^7{3}^6\\ ={\displaystyle \sum _{i=1}^6{\left(-1\right)}^{i+1}{3}^i}\end{array}$

Therefore, the sum of the series $3-9+27-81+243-729$ in the form of sigma notation is ${\displaystyle \sum _{i=1}^6{\left(-1\right)}^{i+1}{3}^i}$ .