A sequence converges if it gets closer and closer to a certain number. Otherwise, it diverges.

In the previous section we looked at the pizza sequence, $\left( \frac{n}{n+1} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$, and noticed that as $\textcolor{#1d4ed8}{n}$ gets larger, the sequence terms get closer and closer to 1. This is called *convergence*, and it's a crucial concept in analysis.

In this lesson we will get a feel for *convergence* (and its opposite, *divergence*) by looking at examples. In the next lesson, we will write a more precise mathematical definition of convergence and begin to prove, rigorously, that sequences converge or diverge.

A *convergent* sequence is one that gets infinitely close to a certain numberβcalled the "limit" of the sequenceβas $\textcolor{#1d4ed8}{n}$ gets bigger. Visually, this means that the terms all get really close and essentially become a horizontal line as $\textcolor{#1d4ed8}{n}$ gets large. (We'll give a more technical definition soon.)

Here are some examples of *convergent* sequences:

$\left( \frac{1}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This is an extremely common converging sequence that is very useful when writing proofs. It converges to 0.

$\left( \frac{(-1)^n}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence alternates positive/negative, but it still converges to 0.

$\left( 5 + \frac{1}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequences converges to 5. Obviously, you could change the number in the sequence to make it converge to anything you want.

$\left( \frac{4n}{n+2} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence converges to 4. If you've ever taken a calculus class, this might feel familiar. Intuitively, the +2 in the denominator essentially becomes meaningless as n gets large, so the n's cancel and we just get 4.

$\left( -3 \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This is the constant sequence -3, and we say that it converges to -3. (This might feel like it goes against the spirit of βgetting closer and closerβ since it's exactly -3 from the very beginning. But when we define convergence technically, you'll see that this counts.)

$\left( \min \{ n, 7 \} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

The min function chooses the smallest option from a set. So if n is small, it chooses n. But if n is big, it chooses 7 instead. This might feel like cheating, but by the technical definition, this sequence definitely converges to 7.

When a sequence converges, we often talk about the *limit*βthe number it converges to. We do that mathematically using this limit notation:

$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{1}{n} \right) = \textcolor{#9333ea}{0}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{(-1)^n}{n} \right) = \textcolor{#9333ea}{0}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( 5 + \frac{1}{n} \right) = \textcolor{#9333ea}{5}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{4n}{n+2} \right) = \textcolor{#9333ea}{4}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( -3 \right) = \textcolor{#9333ea}{-3}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \min \{ n, 7 \} \right) = \textcolor{#9333ea}{7}$

When writing proofs, you might be given some mystery sequence, like $(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$, and be told that it converges to some number, like 2. In this case you would write $\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( x_n \right) = \textcolor{#9333ea}{2}$.

We also sometimes write things like "$\textcolor{#9333ea}{x_n \rightarrow 2}$ as $\textcolor{#1d4ed8}{n \rightarrow \infty}$" , which means "$x_n$ goes to $2$ as $n$ goes to $\infty$." Hopefully that's intuitive enough.

Technically, *divergent* just means "not convergent". It's a pretty boring definition that tells us the sequence will never settle in to a particular limit.

However, we can categorize the ways in which sequences diverge. The first way to diverge is by shooting off to β or -β. We call this *diverging to Β±β*. Here are some examples:

$\left( -n^2 \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence diverges to -β.

$\left( \sqrt n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence diverges to β. (It *looks* like it might converge, but it does not. It keeps getting biggger and bigger, crossing any boundary you set for it.)

$\left( n + \sin (n) \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This wiggly sequence diverges to β, because although it goes up and down, its main trend is always up.

$\left( 5 - |n - 5| \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence starts by going up, but then changes course and goes down forever, diverging to -β.

When a sequence *diverges to Β±β*, we say that its limit is β or -β:

$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( -n^2 \right) = \textcolor{#0d9488}{-\infty}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \sqrt{n} \right) = \textcolor{#0d9488}{\infty}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( n + \sin (n) \right) = \textcolor{#0d9488}{\infty}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( 5 - |n - 5| \right) = \textcolor{#0d9488}{-\infty}$

It's also possible for a sequence to diverge *without* going to Β±β. Sequences like this just wiggle around perpetually, never settling in to a comfortable place. Here are some examples:

$\left( \sin \left(\frac{n}{2}\right) \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence goes up and down forever, never converging to any one number. So it is *divergent*.

$\left( (-1)^n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This sequence alternates between -1 and 1 perpetually, never converging to any one number. So it is *divergent*.

$\left( (-1)^n \cdot n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This one sort of diverges to both β and -β. But since we don't have any special term for that, we just say it *diverges*.

$\left( \lfloor \pi \cdot 10^{n-1} \rfloor \mod 10 \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

This one is extremely dumb. It is the sequence 3, 1, 4, 1, 5, 9... the digits of $\pi$. Since $\pi$ is irrational, its digits never repeat, which means it will always be jumping around and is therefore *divergent*.

Sequences like this are *divergent* but *not* to Β±β. Since these sequences never settle down, they have no limit. That is, the limit does not exist:

$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \sin \left( \frac{n}{2} \right) \right) = \text{D.N.E.}$$\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( (-1)^n \right) = \text{D.N.E.}$

Looking at examples is cute, but if you actually want to learn, you have to practice.

In the game below, you are shown a sequence and asked to classify it as *converging*, *diverging to β*, *diverging to -β*, or *diverging* (not to β or -β). You earn the most points for answering multiple questions in a row correctly, so focus on accuracy first. But once you know what you're doing, you can also earn more points by answering quickly.

Coming soon...