# What is the difference between a sequence and a series in math?

Nov 15, 2015

See explanation

#### Explanation:

A sequence is a function $f : \mathbb{N} \to \mathbb{R}$.
A series is a sequence of sums of terms of a sequence.

For example

${a}_{n} = \frac{1}{n}$ is a sequence, its terms are: 1/2;1/3;1/4;...

This sequence is convergent because ${\lim}_{n \to + \infty} \left(\frac{1}{n}\right) = 0$.

Corresponding series would be:

${b}_{n} = {\Sigma}_{i = 1}^{n} \left(\frac{1}{n}\right)$

We can calculate that:

${b}_{1} = \frac{1}{2}$

${b}_{2} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$

${b}_{3} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{12}$

The series is divergent.