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, , and noticed that as 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 number—called the "limit" of the sequence—as gets bigger. Visually, this means that the terms all get really close and essentially become a horizontal line as gets large. (We'll give a more technical definition soon.)
Here are some examples of convergent sequences:
This is an extremely common converging sequence that is very useful when writing proofs. It converges to 0.
This sequence alternates positive/negative, but it still converges to 0.
This sequences converges to 5. Obviously, you could change the number in the sequence to make it converge to anything you want.
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.
This is the constant sequence -3, and we say that it converges to -3. (This might feel like it goes against the spirit of “getting closer and closer” since it's exactly -3 from the very beginning. But when we define convergence technically, you'll see that this counts.)
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—the number it converges to. We do that mathematically using this limit notation:
When writing proofs, you might be given some mystery sequence, like , and be told that it converges to some number, like 2. In this case you would write .
We also sometimes write things like " as " , which means " goes to as goes to ." 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 ∞ or -∞. We call this diverging to ±∞. Here are some examples:
This sequence diverges to -∞.
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.)
This wiggly sequence diverges to ∞, because although it goes up and down, its main trend is always up.
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 -∞:
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:
This sequence goes up and down forever, never converging to any one number. So it is divergent.
This sequence alternates between -1 and 1 perpetually, never converging to any one number. So it is divergent.
This one sort of diverges to both ∞ and -∞. But since we don't have any special term for that, we just say it diverges.
This one is extremely dumb. It is the sequence 3, 1, 4, 1, 5, 9... the digits of . Since 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:
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.