In mathematics, you don’t understand things. You just get used to them. -Johann von Neumann
Sometimes, I have to deal with series: lots of numbers all added together. Some series clearly approach a limit, like the following:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …
I can visualize this in terms of pies. (What do you want? I’m hungry!) One = one whole pie. So that first number starts me out with one whole pie.
When I add the second number in my series, I’m clearly adding half a pie to that, for a total of a pie and a half.
When I add the third term, I’ve now got one-and-three-quarters pies, and by adding the fourth term, I’ve got one-and-seven-eighths pies.
If I add more and more terms in this series, I’m going to get closer and closer to two whole pies, but I’ll never quite reach it, always being just a tiny sliver away.
And that’s okay! It means that the sum of this series is two. The series has a limit, or — in math-speak — the series converges.
But not all series converge! Take a look at this one:
1 + 1 + 1 + 1 + 1 + 1 + …
The first term starts me out with a pie. Fine, so did the last series. But when I add the second term in, I realize I now have two pies!
In fact, if I add the third one in I have three pies, if I add the fourth in I get four pies, and so on! The more terms I add, the more pies I get.
This last part — that the series doesn’t approach a limit — is really important. It means that I can keep adding more terms, but when I do, the value of the series keeps changing; it doesn’t approach one single value. Therefore, this series does not converge, and the word we give to this is that the series diverges.
Or, to quote Lindsay Lohan (correctly) from Mean Girls, The Limit Does Not Exist!
But what if I were a tease about it? What if I gave you this series:
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + …
After one term, there is a pie, but after two terms, there aren’t any pies; the second term takes the first pie away!
And it looks like the third term adds a pie, and the fourth one takes it away, and this continues. You’re adding pies (one-by-one) and taking them away (one-by-one) as you keep on adding terms.
So you might ask yourself, does the series converge, and if it does, what is its limit?
A mathematician is likely to tell you that the limit does not exist. Why not? Because for half of the terms, the sum is one full pie. But for the other half of the terms, the sum is zero pies. I could group these terms together to show you. For example, if I put my parentheses like so:
( 1 – 1 ) + ( 1 – 1 ) + ( 1 – 1 ) + ( 1 – 1 ) + ( 1 – 1 ) + …
it’s clear that this series sums up to zero. But what if I’m clever, and I organize my terms like this:
1 + ( -1 + 1 ) + ( -1 + 1 ) + ( -1 + 1) + ( -1 + 1) + …
I can see that my series sums up to one whole pie! So who’s right: is it zero pies, one pie, or does the solution simply not exist? To get it right, you have to ask yourself what happens on average.
Well, for half of the terms you have a pie, and for the other half of the terms you have no pies. On average? You have half a pie. Even though at no point do you actually have half-a-pie, this series sums to one-half. (For more rigor, you might want to read this.)
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + … = 1/2.
This is a hard one to wrap your head around, and many mathematicians and physicists that I knew in graduate school were unable to do it. Want to try something more ambitious? Have a go at this one:
1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + …
(Answer here.) And if this isn’t up your alley, don’t worry. We’ll go back to the astrophysics on Wednesday.