Chapter 12.1, Problem 60E

To determine

Find the sum of the given function ${\displaystyle \sum _{j=1}^n{(-1)}^{j+1}{x}^j}$ without using sigma notation.

Answer to Problem 60E

The sum of the given term is $x^1-x^2+x^3+\mathrm{.................}+{(-1)}^{n+1}{x}^n$ .

Explanation of Solution

Given information:

It is given in the question that,

The function is ${\displaystyle \sum _{j=1}^n{(-1)}^{j+1}{x}^j}$ .

By putting the value of $j=1,2,3,4,5,6,7,8,9,10,\mathrm{........}n$ in ${\displaystyle \sum _{j=1}^n{(-1)}^{j+1}{x}^j}$ (summation) the following term is arrived,

  $\Rightarrow {(-1)}^{1+1}{x}^1+{(-1)}^{2+1}{x}^2+{(-1)}^{3+1}{x}^3+\mathrm{.................}+{(-1)}^{n+1}{x}^n$

  $\Rightarrow {(-1)}^2{x}^1+{(-1)}^3{x}^2+{(-1)}^4{x}^3+\mathrm{.................}+{(-1)}^{n+1}{x}^n$

  $\Rightarrow 1x^1-1x^2+1x^3+\mathrm{.................}+{(-1)}^{n+1}{x}^n$

  $\Rightarrow x^1-x^2+x^3+\mathrm{.................}+{(-1)}^{n+1}{x}^n$.