Applications of Riemann Series Theorem

Question

I recently came across Riemann Series Theorem. The theorem seems to be quite general and powerful, making strong statements on the limsup and liminf of rearrangements of conditionally convergent series (specifically that the limsup and liminf can take any arbitrary value). Consequently, I would imagine that it has lots of applications to interesting problems, in proving theorems etc. A cursory search of this site revealed very few such answers and my book does not list any interesting problems.

So, I am looking for problems/theorems which can be easily answered/proved using Riemann series theorem.

Answer 1

There is a nice problem a friend shared with me, which has a short and sweet solution using Riemann Series Theorem.

The problem goes like this :

Show that the set $\mathbb{S}$ of "all bijections from $\mathbb{N}$ to $\mathbb{N}$" has the same cardinality as $\mathbb{R}$.

The solution is below :

The Riemann Series theorem tells us that a conditionally convergent series $\sum_{n=1}^{\infty} a_n$ of real numbers can be rearranged to converge to any given real number $x$. So this gives an injection from $\mathbb{R}$ to $\mathbb{S}$, by simply mapping any real number $x$ to a bijection $\sigma : \mathbb{N} \rightarrow \mathbb{N} $ such that the rearranged sum $\sum_{n=1}^{\infty}a_{\sigma(n)} = x.$ This map is clearly injective, as a convergent series can't have two different sums. Also, we have an injection in the other direction, from $\mathbb{S}$ to $\mathbb{R}$, by mapping a bijection $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ to the real number whose decimal expansion is given by writing, after the decimal point, $\sigma(1)$ many 1's, then $\sigma(2)$ many 0's, then $\sigma(3)$ many 1's, and so on. For example, the identity function in $\mathbb{S}$ would be mapped to the real number $0.1001110000...$ This is an injection because, firstly, two different bijections have different decimal expansions as their images (since given the image, we can exactly reconstruct the bijection), and two different decimal expansions of this form correspond to different real numbers, since each such string after decimal point corresponds to an irrational number (as the string is non-terminating non-recurring, because there are consecutive lists of 1's of arbitrarily long lengths and also the string is clearly not eventually constant), and irrationals have unique decimal expansions. Now that we have an injection in both directions, the Schroder-Bernstein theorem can be invoked to complete the proof that $\mathbb{S}$ and $\mathbb{R}$ are bijective, hence proving that they have the same cardinality.

Answer 2

Let me give you an answer based on a paper by Stewart Galanor that I found very interesting and that I try to summarize here for you (if you need more referenes please ask). My point of view is that the main problems/theorems which can be easily answered/proved using Riemann series theorem are the problems that led to the discovery/invention of this tool. As a result, I think more useful to give a kind of historical answer.

Since that time when Zeno asked about the paradox of Achilles and the tortoise, infinite series have always been a source of wonder. This is mainly due to the fact that series can be manipulated to appear to contradict our understanding of numbers and nature. Zeno’s paradox is still provoking troubles to high school students who study it for the first time.

Mathematicians of the late XVII and XVIII centuries were often puzzled by the results they would get while working with infinite series, and divergent series were condemned very harshly: just to quote Abel: “Divergent series are the invention of the devil,”. Just to quote Kline “By using them, one may draw any conclusion he pleases, and that is why these series have produced so many fallacies and so many paradoxes”

Consider the following example:$$ S=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$

Write out a few terms: $$ \text { (1) } \quad S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}-\frac{1}{12}+\cdots $$ Multiply both sides by 2 : $$ \text { (2) } \quad 2 S=2-1+\frac{2}{3}-\frac{1}{2}+\frac{2}{5}-\frac{1}{3}+\frac{2}{7}-\frac{1}{4}+\frac{2}{9}-\frac{1}{5}+\frac{2}{11}-\frac{1}{6}+\cdots $$ Collect terms with the same denominator (example: $2$ and $-1$,$2/3$ and $-1/3$, $2/5$ and $-1/5$, and so on :

$$ 2 S=2-1+\frac{2}{3}-\frac{1}{2}+\frac{2}{5}-\frac{1}{3}+\frac{2}{7}-\frac{1}{4}+\frac{2}{9}-\frac{1}{5}+\frac{2}{11}-\frac{1}{6}+\cdots $$ We arrive at this:

(4) $\quad 2 S=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots$

We see that on the right side of this equation, we have the same series we started with. In other words, by combining equations 1 and 4, we obtain $$\quad 2 S=S$$ Divide by $S .$ We have shown that $$\quad 2=1.$$

In 1827, Dirichlet discovered this surprising result while working on conditions that ensured the convergence of Fourier series (so you can see this as the first time your question here on MSE made sense :) ). He was the first to notice that it is possible to rearrange the terms of certain series (now known as conditionally convergent series) so that the sum would change. Dirichlet was never able to give an answer of why this is possible. In a paper published in 1837, he did prove that rearranging the terms of an absolutely convergent series does not alter its sum. With the discovery that the sum of a series could be changed, Dirichlet had found the path to follow to prove the convergence of Fourier series (interpret this as an answer to your question). By 1829 he had succeeded in solving one of the preeminent problems of that time.

In 1852, Bernhard Riemann began work on a paper extending Dirichlet’s results on the convergence of Fourier series (Riemann took lots of works that Dirichlet just stated or conjectured without a solution and solved them: an example is the so called Dirichelet principle in calculus of variations, that ensures equivalent formulation for a function to solve a PDE and to be the minimizer of a particular functional). He suspected that divergent series were somehow responsible and soon found a remarkable explanation that accounted for this bizarre behavior, now known as Riemann’s rearrangement theorem, which he incorporated in his paper on Fourier series. Although the paper was completed by the end of 1853, it was not published until after his death in 1866 under the title “On the Representation of a Function by a Trigonometric Series” (and I consider this the main "application" of what we now call Riemann Series Theorem).

Answer 3

This is an addendum to my previous answer to the same question.

It's a natural question to try extending the Riemann Series theorem to a series of functions, with the following question :

Does there exist a series of real-valued functions $\sum_{n=1}^{\infty}f_n(x)$ on some proper (non-empty and not a singleton, hence infinite) interval $I \subseteq \mathbb{R}$ which can be rearranged to get any function $f : I \rightarrow \mathbb{R}$? To be more precise, does there exist a series of real-valued functions as mentioned above such that, given any function $f : I \rightarrow \mathbb{R}$, there exists a bijection $\sigma_f : \mathbb{N}\rightarrow\mathbb{N}$ for which $\sum_{n=1}^{\infty}f_{\sigma_f(n)}(x)$ converges pointwise to $f$ on $I$?

This sounds just like the Riemann Series theorem, but with the series of real numbers replaced by a series of real-valued functions, and instead of rearranging the series of real numbers to get any real number, we are trying to rearrange the series of real-valued functions to get any real-values function $f$ on $I$.

If such a series of real-valued functions existed, then for each $x\in I$, the series $\sum_{n=1}^{\infty}f_n(x)$ would have to be conditionally convergent, because a rearrangement of this is required to converge to $f(x)$ for any real-valued function $f$ on $I$.

So, does such a series of functions exist?

The answer is NO, and to prove this we take the help of the result in my previous answer which was proved using the Riemann Series Theorem in the first place! If such a series of real-valued functions existed, then we would get an injection from the set $\mathbb{F}$ of all real-valued functions on $I$, to $\mathbb{S}$, where $\mathbb{S}$ is as defined in my previous answer, the set of all bijections from $\mathbb{N} \rightarrow \mathbb{N}$. This injection is obtained by simply mapping a function $f \in \mathbb{F}$ to a bijection $\sigma_f \in \mathbb{S}$ such that $\sum_{n=1}^{\infty}f_{\sigma_f(n)}(x)$ converges pointwise to $f$ on $I$. Clearly, this is an injection, because a pointwise convergent series has a unique limit function. Now we know that $\mathbb{S}$ has the same cardinality as $\mathbb{R}$ from my previous answer, but the cardinality of $\mathbb{F}$ is same as that of the power set of $\mathbb{R}$ (see this, and use the fact that $I$ has the same cardinality as $\mathbb{R}$). So, if such a series existed, we would get an injection from the power set of $\mathbb{R}$ to $\mathbb{R}$, which is clearly not possible (because we already have an injection from $\mathbb{R}$ to the power set of $\mathbb{R}$, by mapping $x\in\mathbb{R}$ to $\{x\} \in $ the power set of $\mathbb{R}$. If there was an injection from the power set of $\mathbb{R}$ to $\mathbb{R}$ as well, then we would get a bijection from the power set of $\mathbb{R}$ to $\mathbb{R}$ by Schroder-Bernstein theorem, which contradicts Cantor's theorem.) This shows that no such series of real-valued functions can exist on any proper interval $I\subseteq \mathbb{R}$.