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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As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]