Chapter 12.1, Problem 54E

To determine

Calculate the sum of the equation.

Answer to Problem 54E

The summation of the equation is $-0.69$ .

Explanation of Solution

Given: It is given in the question that the equation is ${\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}$ .

Concept Used:

In this, use the concept of putting the value of variable in the equation and adding it and get the answer and use the concept of nth term of sum of the series.

Calculation: Here, the equation is ${\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}$ .

  $\begin{array}{l}{\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=\frac{{(-1)}^1}{1}+\frac{{(-1)}^2}{2}+\frac{{(-1)}^3}{3}+\mathrm{.........}+\frac{{(-1)}^{99}}{99}+\frac{{(-1)}^{100}}{100}\\ {\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=\frac{(-1)}{1}+\frac{1}{2}+\frac{(-1)}{3}+\mathrm{.........}+\frac{(-1)}{99}+\frac{1}{100}\\ {\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=-\frac{1}{1}+\frac{1}{2}-\frac{1}{3}+\mathrm{.........}-\frac{1}{99}+\frac{1}{100}\\ {\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=-(1+\frac{1}{3}+\frac{1}{5}+\mathrm{.........}+\frac{1}{99})+(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\mathrm{.........}+\frac{1}{100})\\ \Rightarrow (1+\frac{1}{3}+\frac{1}{5}+\mathrm{.........}+\frac{1}{99})=2.94\\ \Rightarrow (\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\mathrm{.........}+\frac{1}{100})=2.25\\ {\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=-2.94+2.25\\ {\displaystyle \sum _{n=1}^{100}\frac{{(-1)}^n}{n}}=-0.69\end{array}$

Conclusion:

The summation is $-0.69$ .