Intro to convergence

How close is arbitrarily close?

Do you guys remember why I write these posts? Cause math is awesome, and it's a ton of fun. You should have fun with it too! The shit mathematicians write in books is sometimes super awesome and cool. I challenge every reader to find a math topic/definition/theorem that they actually enjoy piecing together. Don’t do it because you should, do it because it’s fun.

Yes, it's true, there is a certain unparalleled nerdiness that comes with the "mathematician" label. But we like to get close to things too.

In fact, we so much like to get close to things, that our predecessors have come up with a way to precisely define what it means for some value to get arbitrarily close to another value, but while never touching it.

When a group of values, let’s call them \(x_1, x_2, x_3,\) etc.   "gets close" to another value, which we’ll call \(x\), mathematicians call this convergence. 

An example would be to let \(x_1=4, x_2=4.9, x_3=4.99\), and so on into infinity. The pattern here is easy; at each ensuing index, we add an additional 9 to the end of the number.

The result is that we have a sequence of values that are converging to 5. Loosely speaking, they converge because they get arbitrarily close. They don’t have to ever be equal to 5; it’s enough that they get arbitrarily close!

The point of this post is to talk about the formal definition of convergence (which is the notion of getting arbitrarily close), the one that you’d see in university. but using only words that a high school student can understand. Maybe even an elementary school student, if they're diligent. My first examples are with sequences, as I figure that's a pretty good starting point. 

Let's start turning the ol' thinkin wheels

So that we can easily talk about different terms in our sequence, let’s let \(n\) represent an arbitrary index value. \(x_n\) could represent the first term in the sequence, or the 1,000,000,000th, we don’t know. That will depend on what value \(n\) takes on in the end. What we do know, is that \(x_n\) is indeed one of the terms from the sequence, so it looks like a 4 with a bunch of 9′s after the decimal.    

I want you to take a second to think about what it could mean for a sequence of terms to be arbitrarily close to a value, in this case 5, but never touch it. Specifically, I want you to ask yourself:

As \(n\) gets infinitely large, what’s the smallest distance we can achieve between the terms in \(x_n\) and 5?

In math speak, we write that distance as \(|x_n-5|\). 

To help us out, I’m going to slowly introduce some language for the beginners. Let’s let \(\epsilon\) represent a distance goal for \(|x_n-5|\). In other words, if someone picks an \(\epsilon\) value of \(0.1\), let’s see if we can pick our \(n\) large enough so that the distance between the terms in \(x_n\) are now always less than \(0.1\) away from \(5\). Once we find the index that guarantees we hit our distance goal, we can set that index as a marker, meaning that every term with an index greater than our marker will also reach the distance goal. We label marker indecies with \(N\).

In math speak: Let \(N\) be any index value such that \(|x_n-5|<0.1\) as long as \(n>N\). Then \(N\) is our marker.

You are not allowed to read further until you have either re-read the preceding \(2\) paragraphs 5 times, or you understand it completely!

I think we can reach our distance goal of 0.1, if we let \(n\) be anything larger than \(2\). Let’s check that claim, and then further check the distance for \(n=4\), and \(n=10\).

If \(n=2\), that means \(x_n=4.9\), and then;

\[|x_2-5|=|4.9-5|=|-0.1|=0.1\]

Oh shit, I was WRONG. I was wrong because \(x_2\) is a distance of exactly 0.1 away from \(5\), but I wanted it to be less. Luckily, it’s an easy fix. I’m pretty sure \(x_3\) will work.

It does.

As promised, let’s look at the distances that each of \(x_4,\) and \(x_{10}\) are away from \(5\). \[|x_4-5|=|4.999-5|=|-0.001|=0.001\] \[|x_{10}-5|=|4.999999999-5|=|-0.000000001|=0.000000001\] Everything make sense? If not, don’t be afraid to read it through again.

Something to notice

You know what’s kind of weird? I’m pretty sure that no matter which term I take in my sequence, it will never be equal to \(5\), as there will always be some small positive number I can add to \(x_n\) to make it equal to \(5\).

The second thin to notice is that indeed, no matter how small I make \(\\epsilon\), I can always find an  \(n\) value that makes \(|x_n-5|<\epsilon\).

Believe it or not, we’re essentially done.

The Formal Definition

The formal definition of a sequence on numbers  \(x_n\) converging to some value, \(a\), follows. Know that what you’re about to read is equivalent to the definition you’ll see in university, and it’s using the language that you’re expected to understand. If it confuses you, know that I have fully explained everything in this post barring small notation enhancements. So, if you don’t understand, keep trying. You’ll definitely get there!

A sequence of terms \(x_n\) converges to a value \(a\) if for any and all \(\epsilon>0\), no matter how small, we can find a marker number \(N\) (which is always a natural number) so that \(|x_n-5|<\epsilon\) whenever \(n>N\).

Cool! Sweet job everyone! As usual let me know if there is a part I can explain better! hope you had a rockin’ sweet time!

Cheers,

shel

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