Yeah , we should probably get a better idea of what the heck ‘infinity’ is.
When last we met, we’d gotten up to this point:
and then I did some hand-waving and said something about how those highlighted expressions looked really similar to a ‘geometric series’. A geometric series is a series where there is a constant ratio between each term. For example:
- 1+(1/2)+(1/4)+(1/8)+(1/16)… each term is half of the previous one. And there’s no largest term, you just keep going and going. There’s an infinite amount of terms.
- I realize those don’t look anything alike, but just trust me, it’ll become clear.
- Incidentally, the Fibonacci numbers are a series, but the ratio between each successive term is not constant. But the ratio does converge to a number (phi/golden ratio). Like so:
Anyways, infinity should be a clear concept, but it’s not. When you’re younger, you think of infinity as the biggest number. Ever.
- “You’re a poo-bum-willy-brain!”
- “Well, you’re a stupid-head!”
- “You’re a poo-bum-willy-brain times a thousand.”
- “Well, you’re a stupid-head times a million.”
- “You’re a poo-bum-willy-brain times infinity.”
- “You’re a stupid-head times infinity.”
- “You’re a poo-bum-willy-brain times infinity, plus one!“
- “You’re a stupid-head times infinity, plus a thousand!“
And so forth. It was all going so well, but then they threw infinity into the mix. As it stands, infinity plus one, is infinity. Infinity times two, is infinity. On the flipside to that, though, just because you have an infinite amount of numbers (the infinite amount of terms in a series), it doesn’t mean that adding them up makes infinity. Which means there are two types of series.
Convergence and divergence
A convergent series, when you add up all the terms in it, and no matter how many terms you add, eventually sums up to a number (a limit) which is not infinity. A divergent series has a limit which approaches infinity. Example time, let’s do divergent first, because that’s easy to prove.
The sum of Fibonacci numbers is divergent. Why? Well, each time you’re adding a number larger than the previous number. The number keeps getting bigger, and it gets bigger by a larger amount each time. Simple. Trivial, even.
Let’s check out another series. 1+(1/2)+(1/4)+(1/8)+(1/16)… etc. No matter how many terms you add up, this sum approaches 2, and never goes past two.
I’ll be honest, when I first heard about this, my reaction was “I kinda get it, but if you just kept adding enough numbers, eventually you’d go past two, and eventually you’d get to infinity. Just keep adding numbers! Right? Right?!?” too. But nope, that’s a fact. That series will never go past 2.
You want the proof? You can’t handle the proof!
Actually, you can. First proof is purely intuitive. Go get a ruler. No, seriously, go get a ruler. I’ll wait. You might as well make yourself a cup of tea while you’re at it. Maybe get some crisps or somesuch.
Put your finger at 0 inches. Move it along to 1 inch (the first term in the series). Then you add half, so move it along another half inch, to 1.5 inches. Then moving another quarter inch takes you to 1.75. No matter how many times you do this, all you ever do is move half the previous distance, so you will never quite reach 2. You’ll get closer, and closer, but will always be an infinitesimally tiny distance away. Looking at it another way, from when you add the first ‘1’, all you ever do is halve the remaining distance between 0 and 2. So, the limit of 1+(1/2)+(1/4)… is 2.
Second (more elegant) proof. Relax, it’s a hilariously short proof, considering we’re dealing with an infinite amount of terms. We’ll call this series of terms ‘S’. This is the point where you have to switch from thinking about infinity as ‘the largest number’ and think of it as a process that never ends. Gonna put this in bold, for big-fuck-off emphasis:
Infinity is a verb, not a noun.
That’s important for accepting what’s happening in the right-hand-side terms. Just because the terms I’ve written out aren’t lined up (the last written term in S is 1/16, ,the last in rS is 1/32), it doesn’t mean there’s any trickery going on. Past those terms are more terms, and more and more and it just keeps going and never ends. So don’t worry about how many trailing terms are/aren’t written.
This obviously generalizes to other forms. The one relevant to us is this one:
We already have this:
Which are so tantalizingly close together! So if we can get the expressions in our formula from the form 1/(1+ax) into the form 1/(1-ax), then we express our generating function in terms of the above series and get a closed-form solution! Swish! One last post and we’re home free.