Applications of Riemann Series Theorem 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.
 A: 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).
