Chapter 11.4, Problem 19WE

To determine

To calculate:

The following series with sigma notation:

  $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{....}$

Answer to Problem 19WE

The sigma notation for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{....}$ is ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$ .

Explanation of Solution

Given information:

  $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{....}$

Calculation:

In the given series:

The first term is $1$ . The common ratio or difference between terms is $-\frac{1}{2}$ from the below calculation:

  $\begin{array}{l}d=\frac{1}{2}-1=-\frac{1}{2}\\ d=\frac{1}{3}-\frac{1}{2}=-\frac{1}{2}\\ d=\frac{1}{4}-\frac{1}{3}=-\frac{1}{2}\end{array}$

Now, the sigma notation for the given series is expressed as:

  $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{....}={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$