Playing With Infinite Sums: The Riemann Series Theorem
Welcome to my blog! The first college year is over, and toady i will talk about one of the most fascinating theorems i’ve encountered in this year: the Riemann Series Theorem.
If you aren’t into high level mathematics, most probably you have never heard about this foundamental result, which deals with the rearrangements of infinite sums (called series) and its consequences.
Now you may be probably wondering: “how is this even interesting? if a sum contains infinite (positive) terms, then it always has to go to infinity..!”, but that has only to do with human intuition. In fact, one of the first things we have learnt from the lessons is that certain sequences that go to 0 in the limit of n reaching infinity may converge to a finite value! Some examples of these different behaviors that we shall encounter are:
The first one is a classic example of a geometric series, which consists of sums of powers of a number, and it is known to converge for every value less than 1. The other one is the very famous harmonic series, which is proven to diverge (although veeeery slowly!!) by the integral test…more on that on a future post ;) .
We will now focus on the converging series, and there are two types of it: absolutely and conditionally convergent ( or simply convergent) ones. We have the following definition:
This theorem is called “absolute convergence test”, and (again…) will be proven in a future post. The most notable example of a simply convergent series is the alternating harminic series:
…because its “positive counterpart”, the harmonic series, diverges for what we have seen before. Okay, now we are ready to dive into the heart of this lesson, namely the Riemann Series Theorem, so let’s start with the statement and let’s comment it after:
Now, if you are like me the first time I saw this statement, you would probably be like “what does everything mean!?!”. Don’t worry, it looks scary at first but now I’ll try to make it as straightforward as possible (because it doesn’t need to be hard to understand!). First of all: what the hell is a permutation of the natural numbers? The answer to this question is very simple: is a function from the natural numbers ℕ to itself, and is bijective, meaning that each element of the codomain has exactly one element that gets mapped to it.
For example, we may have a permutation called “f” that maps 1 to 4, then 2 to 7, 3 to 3 and so on, the important thing for f to be a permutation is that there are no two different numbers that get mapped to the same output value (for example, we can’t get another number different from 2 to get mapped to 7). This is the extension of the concept of a finite permutation, that we always do when we shuffle a deck of cards!
Therefore, what the theorem is saying is that, if we take a simply convergent series and we rearrange the terms in some particular way, we may end up with another series that converges to a different value, diverges or even does not have any limit! I know that it can be very counter-intuitive at first, because we are used to the fact that commutativity holds for any finite sum: in fact, if someone will tell you tomorrow that 1+2 yields a different result than 2+1, you would probably think at the very least that something must be wrong with him!
But this is a whole new territory, where infinity enters the game, and all sort of things are possible. So, let’s take a deep breath and jump into the proof:
Keep in mind that we’ve only proven the case when l is a finite number: we can still construct a new series that diverges (or has no limit) following the same tecnique!
That’s it. I hope you have enjoyed this topic and see you in the next episode!
-Leonardo
