## Three Conundrums on Infinity

In this short post, we introduce three conundrums dealing with infinity. This is inspired by my calculus class, as we explore various confusing and confounding aspects of infinity and find that it’s very confusing, sometimes mindbending.

** Order Matters **

Consider the alternating unit series $$ \sum_{n \geq 0} (-1)^n. $$

We want to try to understand its convergence. If we write out the first several terms, it looks like $$ 1 – 1 + 1 – 1 + 1 – 1 + \cdots $$

What if we grouped the terms while we were summing them? Perhaps we should group them like so, $$ (1 – 1) + (1 – 1) + (1 – 1) + \cdots = 0 + 0 + 0 + \cdots $$

so that the sum is very clearly $latex {0}$. Adding infinitely many zeroes certainly gives zero, right?

On the other hand, what if we group the terms like so, $$ 1 + (-1 + 1) + (-1 + 1) + \cdots = 1 + 0 + 0 + \cdots $$

which is very clearly $latex {1}$. After all, adding $latex {1}$ to infinitely many zeroes certainly gives one, right?

A related, perhaps deeper paradox is one we mentioned in class. For conditionally convergent series like the alternating harmonic series $$ \sum_{n = 1}^\infty \frac{(-1)^n}{n}, $$

if we are allowed to rearrange the terms then we can have the series *sum to any number that we want*. This is called the Riemann Series Theorem.

** The Thief and the King **

A very wealthy king keeps gold coins in his vault, but a sneaky thief knows how to get in. Suppose that each day, the king puts two more gold coins into the vault. And each day, the thief takes one gold coin out (so that the king won’t notice that the vault is empty). After infinitely many days, how much gold is left in the vault?

Suppose that the king numbers each coin. So on day 1, the king puts in coins labelled 1 and 2, and on day 2 he puts in coins labelled 3 and 4, and so on. What if the thief steals the odd numbered coin each day? Then at the end of time, the king has all the even coins.

But what if instead, the thief steals from the bottom. So he first steals coin number 1, then number 2, and so on. At the end of time, no coin is left in the vault, since for any number $latex {n}$, the $latex {n}$th coin has been taken by the king.

** Prevalence of Rarity **

When I drove to Providence this morning, the car in front of me had the license place 637RB2. Think about it – out of the approximately $latex {10\cdot10\cdot10\cdot26\cdot 26 \cdot 10 = 6760000}$ possibilities, I happened across this one. Isn’t that amazing! How could something so rare happen to me?

Amazingly, something just as rare happened last time I drove to Providence too!