What does it mean for a sequence to converge or diverge?

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, (nn+1)n∈N\left( \frac{n}{n+1} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}, and noticed that as n\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.

What does convergent mean?

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

Here are some examples of convergent sequences:

(1n)n∈N\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.

((βˆ’1)nn)n∈N\left( \frac{(-1)^n}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

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

(5+1n)n∈N\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.

(4nn+2)n∈N\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.

(βˆ’3)n∈N\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 β€œgetting closer and closer” since it's exactly -3 from the very beginning. But when we define convergence technically, you'll see that this counts.)

(min⁑{n,7})n∈N\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 limitβ€”the number it converges to. We do that mathematically using this limit notation:

lim⁑nβ†’βˆž(1n)=0\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{1}{n} \right) = \textcolor{#9333ea}{0}lim⁑nβ†’βˆž((βˆ’1)nn)=0\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{(-1)^n}{n} \right) = \textcolor{#9333ea}{0}lim⁑nβ†’βˆž(5+1n)=5\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( 5 + \frac{1}{n} \right) = \textcolor{#9333ea}{5}lim⁑nβ†’βˆž(4nn+2)=4\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \frac{4n}{n+2} \right) = \textcolor{#9333ea}{4}lim⁑nβ†’βˆž(βˆ’3)=βˆ’3\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( -3 \right) = \textcolor{#9333ea}{-3}lim⁑nβ†’βˆž(min⁑{n,7})=7\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 (xn)n∈N(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⁑nβ†’βˆž(xn)=2\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( x_n \right) = \textcolor{#9333ea}{2}.

We also sometimes write things like "xnβ†’2\textcolor{#9333ea}{x_n \rightarrow 2} as nβ†’βˆž\textcolor{#1d4ed8}{n \rightarrow \infty}" , which means "xnx_n goes to 22 as nn goes to ∞\infty." Hopefully that's intuitive enough.

What does divergent mean?

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 ∞ or -∞. We call this diverging to ±∞. Here are some examples:

(βˆ’n2)n∈N\left( -n^2 \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

This sequence diverges to -∞.

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

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

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

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

(5βˆ’βˆ£nβˆ’5∣)n∈N\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 -∞.

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

lim⁑nβ†’βˆž(βˆ’n2)=βˆ’βˆž\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( -n^2 \right) = \textcolor{#0d9488}{-\infty}lim⁑nβ†’βˆž(n)=∞\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \sqrt{n} \right) = \textcolor{#0d9488}{\infty}lim⁑nβ†’βˆž(n+sin⁑(n))=∞\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( n + \sin (n) \right) = \textcolor{#0d9488}{\infty}lim⁑nβ†’βˆž(5βˆ’βˆ£nβˆ’5∣)=βˆ’βˆž\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 ±∞. Sequences like this just wiggle around perpetually, never settling in to a comfortable place. Here are some examples:

(sin⁑(n2))n∈N\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.

((βˆ’1)n)n∈N\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.

((βˆ’1)nβ‹…n)n∈N\left( (-1)^n \cdot n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

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

(βŒŠΟ€β‹…10nβˆ’1βŒ‹mod  10)n∈N\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 Ο€\pi. Since Ο€\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 ±∞. Since these sequences never settle down, they have no limit. That is, the limit does not exist:

lim⁑nβ†’βˆž(sin⁑(n2))=D.N.E.\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( \sin \left( \frac{n}{2} \right) \right) = \text{D.N.E.}lim⁑nβ†’βˆž((βˆ’1)n)=D.N.E.\lim_{\textcolor{#1d4ed8}{n \rightarrow \infty}} \left( (-1)^n \right) = \text{D.N.E.}

The Sequence Convergence Game

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 ∞, diverging to -∞, or diverging (not to ∞ or -∞). 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...