Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody’s answering this question so I’ll try it here. This is really a reference request: Has a certain kind of proof ever been used? A series ∑nan converges absolutely if ∑n|an|<∞. It converges unconditionally if it converges to a finite number and all of its rearrangements converge to that same number. For series of real … Read more

Asymptotic behavior of sums with binomial coefficients

Let f(n) be defined on non-negative integers and define S(m) = \sum_{n=0}^{m} \binom{m}{n} f(n). Depending on the choice of f, S(m) may have a closed form; for example, when f(n) = n, then S(m) = m 2^{m-1}. My question is, does a general theory exist which relates the asymptotic behaviors of f and S, and … Read more

Is there any closed form expression for $\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$?

Is there any closed form expression for the following serie? $$\sum_{k=2}^\infty(-1)^k \left(- \frac{1}{2}\right)^{\frac{k(k+1)}{2}}$$ Or at least a proof that it is an irrational number. The context of this problem is given by the following link: https://math.stackexchange.com/questions/2270730/whats-the-limit-of-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2-sqrt2 In which it is proposed the problem of finding a closed form for the following nested radical: $$R = … Read more

Non-singularity of a series of matrices

Let A1, A2 be n×n real matrices. Suppose that A1 and A2 are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let B1, B2 be two n×m real matrices of full column rank (i.e. rankB1=rankB2=m) and define R1:=[B1|A1B1|A21B1|⋯|An−11B1] R2:=[B2|A2B2|A22B2|⋯|An−12B2] (In control theory the above-defined matrices are called reachability … Read more

Convergence of an=(1−12)(12−13)…(1n−1n+1)a_n=(1-\frac12)^{(\frac12-\frac13)^{…^{(\frac{1}{n}-\frac{1}{n+1})}}} [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 3 years ago. Improve this question I’m interesting to see the opinion of MO about my question which I posted here in SE, Answers I received have not convinced me, … Read more

Counting “motifs” with the same “energy”

This question is motivated by physics — trying to understanding the so-called ‘accidental’ (i.e. non representation-theoretic) degeneracies that occur in the spectrum of the Haldane–Shastry spin chain — but let me formulate it combinatorially. The resulting introduction is still rather long, my apologies. Remark. I believe that the terminology “motif” (Haldane et al, Phys. Rev. … Read more

Limit of the real part of a geometric sequence

I came across the following problem, which turned out to be surprisingly hard: Show that lim Intuitively, setting z=\frac{1+i\sqrt{7}}{2}=\sqrt{2}e^{i\theta}, we see that |\mathrm{Re}(z^n)|=2^{\frac{n}{2}}|\cos (n\theta)|, so that as long as n\theta \ (\mathrm{mod}\ \pi) does not approach \frac{\pi}{2} exponentially fast on subsequences, |\mathrm{Re}(z^n)| should go to \infty. But how to prove that ? Here is a … Read more

When is \lfloor C^n \rfloor \mod b\lfloor C^n \rfloor \mod b efficiently computable?

For real irrational C > 1 and natural n,b, define a(C,n,b)=\lfloor C^n \rfloor \mod b Q1 For which C,b is a(C,n,b) computable in time polynomial in \log{n}? Searching in OEIS suggests that for C \in \{1+\sqrt{2},1+\sqrt{3},(1+\sqrt{5})/2\}, a(C,n,b) satisfy linear recurrence with constant coefficients and so it is efficiently computable over the integers and all bases … Read more

Square root of a sequence given by a linear recurrence relation

This question is closely related to this one but in a sense a converse of it. Let us concentrate for simplicity on a third order relation $x_{n+3}=ax_{n+2}+bx_{n+1}+cx_n$ with given intial terms $x_1,x_2,x_3$ and assume $x_n\geq0$ for all $n$. Assume we fix $a,b,c\in\mathbb{R}$ generically and vary $x_1,x_2,x_3\in\mathbb{R}$. We are looking for a second order recurrence $y_{n+2}=ry_{n+1}+sy_n$ … Read more

What fraction of fractions does Cantor’s famous sequence enumerate?

Cantor’s famous sequence 11,12,21,13,31,14,23,32,41,15,51,16,… provides a bijection between natural numbers and positive rational numbers or cancelled fractions. About half of the fractions qi lie within 0<x≤1. What is the limit of the ratio limk→∞|{x∈R|n<x≤n+1}∩{q1,q2,…,qk}||{x∈R|0<x≤1}∩{q1,q2,…,qk}| for n \in \mathbb{N}? Is there an n for which the limit is 0? And if so, what is the first … Read more