Chapter 9, Problem 1RE

What is the difference between an infinite sequence and an infinite series?

To determine

The difference between an infinite sequence and an infinite series.

Answer to Problem 1RE

The sequence is $a_1,a_2,a_3,\dots ,a_{\left(\infty \right)}$ and $a_n=b_1+b_2+\cdots +b_{\left(\infty \right)}$ is a series.

Explanation of Solution

An infinite sequence is simply the numbers written in sequence separated by a comma. An infinite sequence is simply the sequence of numbers up to $\infty$ .

$a_1,a_2,a_3,\dots ,a_{\left(\infty \right)}$

An infinite series is the sum of numbers upto infinity. Infinite series is a sub-part of an infinite sequence.

$a_n=b_1+b_2+\cdots +b_{\left(\infty \right)}$

Consider the example.

The sequence is $\frac{1}{3},-\frac{2}{5},\frac{3}{7},-\frac{4}{9},\mathrm{...}$ and the series is $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}$ .

Hence, the sequence is $a_1,a_2,a_3,\dots ,a_{\left(\infty \right)}$ and $a_n=b_1+b_2+\cdots +b_{\left(\infty \right)}$ is a series.