Chapter 11.4, Problem 26WE

To determine

To calculate:

The following series with sigma notation:

  $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{....}$

Answer to Problem 26WE

The sigma notation for $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{....}$ is ${\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{\left(2n+1\right)}}$ .

Explanation of Solution

Given information:

  $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{....}$

Calculation:

In the given series:

The first term is $1$ . Here, each term appears as $\left(-1\right)$ divided by $2n+1$ .

Therefore the summand includes $\frac{{\left(-1\right)}^n}{\left(2n+1\right)}$

The value of the lower limit is calculated by using the last term as:

  $\begin{array}{l}\frac{{\left(-1\right)}^n}{\left(2n+1\right)}=1\\ \frac{{\left(-1\right)}^n}{\left(2n+1\right)}=\frac{1^0}{2\left(0\right)+1}\\ n=0\end{array}$

So, the lower and upper limit is $0$ and $\infty$ . Now, the sigma notation for the given series is expressed as:

  $1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\mathrm{....}={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{\left(2n+1\right)}}$