10.1 Defining Convergent and Divergent Infinite Series

A sequence is an ordered list of numbers, usually written as ${a_1, a_2, a_3, \dots}$, where each term depends on its position n. You can define a sequence explicitly, such as ${a_n=\frac{1}{n}}$, or recursively, where each term depends on earlier ones. A series is the sum of the terms of a sequence, written as ${\sum_{n=1}^{\infty} a_n}$ for an infinite series. In Unit 10, the main question is whether the partial sums of a series approach a finite number. If they do, the series converges; if they do not, the series diverges.

You should separate the ideas of a sequence and a series clearly. A sequence is the list of terms itself, while a series is what happens when you add them. To study an infinite series, you look at its partial sums: ${S_1=a_1, S_2=a_1+a_2, S_3=a_1+a_2+a_3}$, and so on. If the sequence of partial sums approaches a finite limit, then that limit is the sum of the infinite series. So series are really limit problems about sums.

Example 1: The sequence ${a_n=\frac{1}{n}}$ gives 1,${\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots}$ This is a sequence because you are listing terms in order.