Chapter 12.1, Problem 68E

To determine

Find the sum by using sigma notation.

Answer to Problem 68E

The sum by the sigma notation is ${\displaystyle \sum _{k=0}^{99}{(-1)}^k(k+1)x^k}$ .

Explanation of Solution

Given:

It is given in the question that the sum of the few terms are $1-2x+3x^2-4x^3+5x^4\mathrm{............}-100x^{99}$ .

Concept Used:

In this, use the concept of understand the pattern of the term and write by using sigma notation.

Calculation:

Here, the term is $1-2x+3x^2-4x^3+5x^4\mathrm{............}-100x^{99}$ .

It is seen in the term that k varies from $0to99$ for the expression ${(-1)}^k(k+1)x^k$ .

So, there are many possible answers. A natural way to write is ${\displaystyle \sum _{k=0}^{99}{(-1)}^k(k+1)x^k}$ .

Conclusion:

The sum by the sigma notation is ${\displaystyle \sum _{k=0}^{99}{(-1)}^k(k+1)x^k}$ .