@Secret that's interesting to check. However, these days I'm struggling hard to calculate some series by series manipulations rather than by using integrals. In this very moment it still seems very hard to do that, perhaps I miss something in the picture.
@Waiting There are numerical methods on infinite series, but I am not very familar with the algorithms yet. Perhaps a numerical analysis on partial sums or portions of the series might give some idea on what special functions might be related to them or what is the most natural way to split them
@Secret These series (well, some alternating series) seem very hard to approach by other means. Actually they could be attacked with Polylogarithms, but the calculations are far from being any elegant, maybe long and tedious tasks.
@Waiting If they oscillate very rapidly and wildly, then I agree that it may be ill-formed for numerical treatments as it will be hard to prevent the error from blowing up quickly due to floating point arithmetic cancellations
@Secret The aim is to calculate them in closed-form which I already did that, but I'm not able to finish them by series manipulations only. I need to do more investigations.
If it had not work for a very long time, there is a slim possibility that those series might happened to form a structure with some associated no-go theorem, often when that happens I will start to try to prove the no-go theorem and see if it is the case
we are knew of no-go theorems like the unsovability of quintics in radicals and nonelementary integrals and so on. I will nto be suprised if some infinite alternating series under the operation of series manipulation actualy form some ring like structure with its associated no-go theorems
though in either case it is still kinda suprising since alternating series tend to behave nicer than their non alternating counterparts
@Secret hmmm, I need to ponder more over this matter before issuing even an opinion. In general, turning integrals into series it is one of the best ways to go for a solution.
Intuition says it is possible to do them by series manipulations. Some mystery lies around those series which simply make them more attractive.
If you take @Akiva's function and modify it so that it's always 0 and goes up to 1 in that points with a triangle of decresing basis does this result hold?
"For any $\epsilon$ there exists an $N$ such that for all $m,n>N$ we have $\frac1{f(m)}-\frac1{f(n)}<\epsilon$" is equivalent to saying the limit to infinity exists, right?
> 6. Describe a topic, idea, or concept you find so engaging that it makes you lose all track of time. Why does it captivate you? What or who do you turn to when you want to learn more?
@AkivaWeinberger I'm kinda in the same boat. One of my prompts: Tell us about a concept, theory, or topic you have explored simply because it sparked your intellectual curiosity. Why do you find it intriguing? How do you want to further explore it?
Let $\alpha \in \mathbb{R}$\ $\mathbb{Q}$, and we denote $a_n=\{n\alpha\}$, where $\{x\}$ is the fractional part of $x$. Calculate $$\lim_{n\to\infty} (a_2-a_1)(a_3-a_2)\cdots (a_{n+1}-a_n)$$
You know the classic puzzle about taking a chessboard, cutting off two opposite corner squares, and trying to tile the remaining 62 squares with dominoes?
