I just learned from the textbook that apparently whether the series

converges is still open, which I find rather surprising. The reference the book lists is the book Mazes for the Mind by Clifford Pickover, St. Martin Press, NY, from 1992, but Dr. Pickover has informed me that he believes the problem is still unresolved; he also discusses it in his book The Mathematics of Oz, Cambridge University Press, 2002. I would be very curious to hear from updates or suggestions, if you have any.

Here is a slightly technical (and very quick, and not particularly deep) observation: The issue seems to be to quantify how small is, when it is small, or more precisely, how sparse the set of values of is for which the sine function is “significantly small.” One could start by looking at so that is small for some , so we are led to consider the standard convergent approximations to , satisfying . This means that is close to, but slightly larger than, and so the question leads us to the problem of how sparse the sequence of numerators of the rational approximations to actually is, something about which I don’t know of any results.

Below I display some graphs for the partial sums of the series. Let . The first graph shows vs. for . In the other graphs, goes up to 300, 1000, and 100000. (Thanks to Richard Ketchersid for the code.) It is not clear to me that the last graph is accurate or that it allows us to draw any conclusions (it certainly seems to suggest that the series converges to a number slightly larger than 30); it may well be that further jumps are beyond the range I chose, or that the approximations Maple uses in its computations are not fine enough to examine very large values of the series.

Notice that examination of just the first few values of would suggest that the series converges to a number near 4.8. In fact, for many “natural” series, the 300-th partial sum gives an accurate approximation of their value. However, as the third graph reveals, a jump suddenly occurs, slightly after we pass the 350-th partial sum. The jump occurs at notice that 355 is very close to an integer multiple of , in fact

Even though the fourth graph does not reveal any further jumps, it is not clear that they won’t occur at certain values of past the 10000 mark.

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Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]