*Infinitely many mathematicians walk into a bar. The first one walks up to the bartender and says “I’ll have a pint.” The second one says “I’ll have half a pint.” The third one points to the second mathematician and says “I’ll have a quarter of a pint”, and so on. The bartender says “You’re all bastards” and pulls out two pints.*

One of the biggest obstacles in studying Calculus for the first time is grappling with the possibility that you can add together infinitely many positive numbers and not get infinity. Today I want to talk about the most basic sorts of these cases. In this post, intended for the non-mathematician, I’ll talk about geometric series by dissecting the joke above.

The first mathematician orders pint, the second mathematician orders a pint, the third mathematician orders of a pint, and so on. At this rate, the -th mathematician orders of a pint. To grapple with the total amount of beer ordered by all of the mathematicians, which we’ll call , let’s first calculate how much total beer is ordered after the -th mathematician has ordered. Let’s call this quantity .

If we multiply both sides by and distribute, we see that

Now the portion inside the parentheses is just (check this!), and so

Solving for , we see that

Since denotes the total amount of beer ordered by the first mathematicians – to determine how much beer is consumed by all of the mathematicians, all we need to determine is what the sequence of numbers tends to as tends to infinity. Notice that when gets larger and larger, the term gets smaller and smaller – and tends to zero. So, using the notation adopted above,

Thus, collectively, all of the mathematicians ordered pints.

The infinite sum, or “series”, we constructed above is called a geometric series. Why? Because the ratios of successive terms is constant:

- ,
- ,
- , and so on.

Geometric series are the easiest series to deal with because, using the same methods above, they’re very easy to calculate. Indeed, if , we have that

To test your understanding – try to replicate the argument used to explain the joke above to prove this [above we took ]. Also, try to see why the argument fails when or .

That’s about it for geometric series. The inspiration for this post came from telling this joke in my calculus class today and none of my students getting it at first. All of them are assigned to read this post. For those mathematicians who stuck around until the end:

*Infinitely many mathematicians walk into a bar. The zeroeth one walks up and says “I’ll have a pint. The first one walks up to the bartender and says “I’ll have a pint too.” The second one says “I’ll have half of what he had” The third one points to the second mathematician and says “I’ll have a third of what the second guy had”, and so on. The bartender says “You’re all bastards” and pulls out pints. Classic. *

Infinitely many mathematicians walk into a bar. The zeroeth says “I’ll have a pint.” The first says, “I’ll have pints.” The second says “I’ll have pints.” The third says “I’ll have pints$, and so on. The bartender says “You’re all bastards,” and pulls out a pint.

Hahahaha.

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