Some series are more special than others: The Riemann Series Theorem (2)
Hello! Welcome to this new article about the Riemann Series Theorem. In the last episode we have seen that a rearrangement of the terms of any simply convergent series may change the value it converges to, make it diverge or even not have any limit. Now we will set aside those for some time, and focus on some other series that have the opposite property, namely that no matter which permutation we choose, we will always end up with the same behavior (or the same limit, when it is finite!). This is sometimes called the commutative property of series, and it is very difficult to spot at first. For instance, look at these two similar-looking series:
Now, imagine you know nothing at all about the Theorem we have seen on the last episode, would you believe me if I said that only the second one, no matter which permutation you choose to apply, will always converge to the same value?
Yeah, that seems pretty odd at first, but don’t worry, once we’re done it will all make sense. So without any time to lose, let’s jump right into today’s biggest
Proof: By the hypothesis, it must be that a_n ---> 0 in the limit as n goes to infinity. Moreover, the "remainder series":
which is a direct result of the terms on the right converging both to S. Now we do the following:
first of all, we fix ε > 0
we then know that there exists N’ natural number such that for all numbers greater than it:
we also know that, since the series converges
Now given a permutation sigma, there must exist i_1,…,i_N’ indices such that
σ(i_1)= 1
…
σ(i_N’) = N’
this is because a permutation is a bijective function.
okay, now here’s the trick: if we take K > max{N’’, i_1,…i_N’} we have that
we now make the following observations:
the second observation follows comes from the fact that any n which is not one of the indeces we talked about before, gets necessarily mapped by sigma into a number strictly greater than N’. But now note that
which is less than ε by hypothesis. Therefore we can conclude that
But remember that our choice of ε is arbitrary, so this is the definition of
which is what we wanted to prove.
And here it is, I hope you’ve appreciated this beautiful proof. In the next articles i will provide you with some visual examples to better understand this topic. See you soon!
-Leonardo
