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.