Chapter 12.1, Problem 74E

To determine

To find: The sum of the given expression with sigma notation.

Answer to Problem 74E

The sum of the given expression is ${\displaystyle \sum _{n=1}^{100}{\left(-1\right)}^{n+1}n{x}^{n-1}}$.

Explanation of Solution

Function used:

The function $\sum$ is defined as,

$a_1+a_2+a_3+a_4\mathrm{...}+a_{n-1}+a_n={\displaystyle \sum _{k=1}^n{a}_k}$ (1)

Given:

The given sequence is,

$1-2x+3x^2-4x^3+5x^4\cdots -100x^{99}$

Calculation:

In the given sequence both the sign and power are changing.

The power value changes linearly and using equation (1) the generated pattern will be,

$1-2x+3x^2-4x^3+5x^4\cdots -100x^{99}={\displaystyle \sum _{n=1}^{100}{\left(-1\right)}^{n+1}n{x}^{n-1}}$

Thus, the sum of the given expression is ${\displaystyle \sum _{n=1}^{100}{\left(-1\right)}^{n+1}n{x}^{n-1}}$.