Intuiting Sequences

by Justin Skycak (@justinskycak) on October 05, 2017

A sequence is a list of numbers that has some pattern.

This post is part of the series An Intuitive Primer on Calculus.

Want to get notified about new posts? Join the mailing list and follow on X/Twitter.

A sequence is a list of numbers that has some pattern. For example, $2, 4, 6, 8, \ldots$ is the sequence of even numbers, and $1, 3, 5, 7, \ldots$ is the sequence of odd numbers.

We can interpret sequences as functions evaluated at whole numbers. For example, the sequence of even numbers is like a function $y=2x,$ where $x$ is evaluated at $1, 2, 3, 4, \ldots.$ The result is $2(1), 2(2), 2(3), 2(4), \ldots$ which simplifies to $2, 4, 6, 8, \ldots.$

Because we’re only plugging in whole numbers for $x,$ we usually use the term $n$ instead of $x,$ and $a_n$ instead of $y.$ So the sequence of even numbers is given by the rule $a_n = 2n,$ as follows:

$\begin{align*} a_1 &= 2(1) = 2 \\[5pt] a_2 &= 2(2) = 4 \\[5pt] a_3 &= 2(3) = 6 \\[5pt] a_4 &= 2(4) = 8 \\[5pt] \vdots \end{align*}$

To check your understanding, observe that the sequence of odd numbers is given by $a_n = 2n-1{:}$

$\begin{align*} a_1 &= 2(1) - 1 = 2 - 1 = 1 \\[5pt] a_2 &= 2(2) - 1 = 4 - 1 = 3 \\[5pt] a_3 &= 2(3) - 1 = 6 - 1 = 5 \\[5pt] a_4 &= 2(4) - 1 = 8 - 1 = 7 \\[5pt] \vdots \end{align*}$

The sequences of odd and even numbers keep getting bigger as they go on, but not all sequences are like that. For example, consider the sequence $a_n = \frac{1}{n}.$ This sequence, which starts off $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ is a sequence of fractions that keeps on getting smaller.

Not only does this sequence get smaller, but it also gets closer and closer to a single number as the sequence goes on. That number is zero – we say that the sequence converges to zero.

Likewise, consider the sequence $a_n=\frac{n-1}{n}.$ This sequence is an increasing sequence that gets closer and closer to $1,$ so we say it converges to $1.$

You can usually tell whether a sequence converges by thinking about what happens when you plug in very large numbers for $n.$ For example, the sequence $a_n=\frac{1}{n}$ converges to $0,$ and plugging in a million for $n$ gives

$\begin{align*} a_{1000000}=\frac{1}{1000000}=0.000001, \end{align*}$

which is really close to $0.$ Likewise, the sequence $a_n=\frac{n-1}{n}$ converges to $1,$ and plugging in a million for $n$ gives

$\begin{align*} a_{1000000}=\frac{99999}{1000000}=0.99999, \end{align*}$

which is really close to $1.$

This post is part of the series An Intuitive Primer on Calculus.

Want to get notified about new posts? Join the mailing list and follow on X/Twitter.