Chapter 11.4, Problem 13WE

To determine

To calculate:

Given series using sigma notation.

Answer to Problem 13WE

Given series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}$ using sigma notation is ${\displaystyle \sum _{n=1}^{99}\frac{n}{n+1}}$ .

Explanation of Solution

Give information:

Expression:

  $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}$

It is given that,

  $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}$

Need to rewrite the series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}$ using sigma notation,

Rewriting the given series by $\frac{n}{n+1}$ ,

Representation of the series by sigma notation by taking index $n=1$ to $n=99$ as

  $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}={\displaystyle \sum _{n=1}^{99}\frac{n}{n+1}}$

Therefore, Given series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\mathrm{....}+\frac{99}{100}$ using sigma notation is ${\displaystyle \sum _{n=1}^{99}\frac{n}{n+1}}$ .