In the last lesson we said that a sequence converges if its terms get "infinitely close" to some limit value. But what exactly does that mean?
A sequence converges if it can get as close as you want to the limit value, just by scrolling along the sequence to a larger value.
For example, if you want to get within ±0.1 of its limit value, 0, you just have to scroll to . And if you want it to get within ±0.01 of its limit, you have to scroll to . No matter how close you want to get to the limit, there's a place where it happens.
We can visualize this by drawing a tube of closeness and waiting for the sequence terms to enter the tube. Try making this tube bigger or smaller and watch what happens:
No matter how small the tube gets, the terms of a converging sequence will eventually go inside.
Here's an example of a sequence that—surprisingly—does NOT converge to 0 (it is diverging):
When you first look, it certainly appears to converge to 0! But in fact, as you scroll farther, you can see that it never stays in the tube. It always leaves eventually.
This is how we know that the sequence does NOT converge to 0.
A converging sequence, on the other hand, will become a permanent resident eventually. This one is undecided at first, but after a while it enters the tube permanently:
Alright... Time to describe, in a sentence, what it means to be convergent.
Below are some fridge magnets that you can use to build the definition. Drag them into place and then click the button to check your answer:
Amazing! You've successfully built a definition of sequence convergence. But our current tube terminology—rad as it is—doesn't really lend itself to writing proofs.
Let's rewrite our definition to be more mathematical, starting by defining some variables:
Often we make be the very first point where the terms enter the tube. But it doesn't have to be that way! can be any point beyond which all the terms are in the tube.
You're allowed to choose an extra big "just to be safe" if it makes you happy.
Here's the interactive tube graph with and labeled:
Now that we have these two variables, we can write the definition of convergence more precisely. As a reminder, here's the definition in tube terms:
A sequence converges to if... for any tube centered at , no matter how small, there is an index beyond which all the sequence terms are inside the tube.
Now, fill in the equivalent math terms using the fridge magnets below:
|Tube terms||Math terms||Quantifiers|
|For any tube centered at , no matter how small...|
|there is an index...|
|beyond which all the sequence terms...|
|are inside the tube.|
So our final definition is...
A sequence converges to if for all , there exists some such that for all , .
In the next lesson, we'll use this definition to prove that a specific sequence is convergent.