Chapter 12.1, Problem 71E

Sigma Notation Write the sum using sigma notation.

71. $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{999\cdot 1000}$

To determine

Sum of the given expression with sigma notation.

Answer to Problem 71E

The sum of the given expression is ${\displaystyle \sum _{i=1}^{99}\frac{1}{n\left(n+1\right)}}$.

Explanation of Solution

Given:

The given sequence is,

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}\cdots +\frac{1}{999\times 1000}$

Function used:

The function $\sum$ is defined as,

$a_1+a_2+a_3+a_4\mathrm{...}+a_{n-1}+a_n={\displaystyle \sum _{k=1}^n{a}_k}$ (1)

Calculation:

The given sequence is,

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}\cdots +\frac{1}{999\times 1000}$

In above sequence there is a pattern of value multiplied immediately to its successor value up to 100, therefore using function of equation (1) the generated pattern be,

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}\cdots +\frac{1}{999\times 1000}={\displaystyle \sum _{i=1}^{99}\frac{1}{n\left(n+1\right)}}$

Thus the sum of the given expression is ${\displaystyle \sum _{i=1}^{99}\frac{1}{n\left(n+1\right)}}$.