Rolfe Schmidt

When we were walking back from the playground on Saturday, Gunnar asked me

“If I take one half, plus one third, plus one fifth, plus for all of the prime numbers, will that make one?”

What a question! The answer is no, when you add it up you’ll find out that the sum just keeps growing and eventually is bigger than any number you can think of. It is infinite. I told him that. He said

“Do you mean it gets bigger than a googolplex? a googolplex googolplexes?”

Yes and yes. And bigger still. I just couldn’t think of an easy way to explain why I know this. In my next post, I’m going to give the simplest argument I know. As a warm-up, I’m going to talk a little bit about a simpler problem.

Some sums are finite: Geometric Series

I didn’t try to explain to Gunnar why is infinite, but I did want to make sure he knew it was a really good question. Sometimes you can add an infinite number of terms and get a finite result. To convince him, I had him stop at the next street corner.

There wasn’t any traffic around, so I said “Let’s just cross halfway”.

Once we were in the middle of the street, I said “OK, that’s half. Now how much is half of what is left?”

“A fourth.” said Gunnar.

“OK, let’s cross a fourth more. There, that’s a half plus a fourth. Now how much is half of what is left?”

“An eighth!” said Gunnar, starting to get it.

“OK, let’s cross an eighth more. There, that’s half plus a fourth plus an eighth.”

And so we continued until we only had a sixty-fourth of the street left. To Zeno’s dismay, we just stepped up onto the sidewalk after that. Gunnar got the point: the process would never stop, but once we added all the terms we would have crossed exactly one street. .

This is an example of a geometric series. To get a general geometric series, pick any number . Now we can look at the sum

If then the series is , which obviously just keeps getting bigger. If , the problem just gets worse. (Excercise: whate happens when ?)

But if then this sum will get closer and closer to a number that we will call the “sum of the series”. We can figure out exactly what this number is. If we call it then:

So , or

All these long sums get tedious to write, so we will start using some shorthand. We will use a Greek capital Sigma, , to denote a sum, like this:

(Remember, ! ) For an infinite series, we’ll need to write

It looks fancy, but it’s just a way to save some writing and make things more precise. Whenever I put those “…” marks in, I was trusting that you’d know what was supposed to come next. That will cause trouble down the road, and our new notation avoids it. Have a look here if you want a bit more detail.

So let’s get back to our street crossing experiment. At each step, we were going half as far as we had in the previous step. So . As we crossed the street, we crossed half, then a fourth, then and eighth, and so on. So we can write the number of streets we crossed like this:

This almost looks like the geometric series in our formula above, but it starts with one-half, not one. We’ll fix that by adding one to both sides:

So

Whew! I’d have been a little worried otherwise.

When I write about prime numbers in my next post, we’ll use these geometric series a lot. It’s amazing how many places they pop up.