37 votes

What is the limit of $a (n + 1) / a (n)$?

The value is close to $e$ but not. It's actually the positive real root of $p(t) := t^3 - 2t^2 + t - 8$. This is solvable via ACSV (see book by Pemantle and Wilson 2013). To summarize, the ...
33 votes

Behavior of $n^\alpha \sin^{\circ\, n}(n^{-\alpha}x)$

A rescaling is needed for a nontrivial limit. As discussed in Iteration of Sine and Related Power Series by C. Towse (2014), denoting the $n$-th iterate by $\sin^{\circ n}x$, one has the limit $$\lim_{...
31 votes
Accepted

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

Update. A research collaboration growing out of this question and some of its answers has now resulted in the following article, providing an account of the rearrangement number: A. Blass, J. ...
25 votes
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Convergence of Fourier series

Convergence in $L^p$, $p>1$. True, by M. Riesz's Theorem. This is a standard topic in every harmonic analysis course, with several readable proofs. Convergence pointwise almost everywhere, $p>...
25 votes
Accepted

Fraction of $S_n$ reachable by using every transposition once as $n\to\infty$?

The value is $1/2$. The problem can be reformulated as follows: How to sort a non-sorted list $(a_1,...,a_n)$ that is a permutation of $\{1 .. .n\}$ with the parity the same as $n \choose 2$ by ...
  • 8,688
24 votes
Accepted

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

The product does not tend to the limit zero. For any irrational number $\alpha$ one can show that $$ \limsup_{N\to \infty} \prod_{n=1}^{N} |1- e(n\alpha)| = \infty. \tag{1} $$ (Here I use the usual ...
  • 42.7k
23 votes
Accepted

Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$

This is exactly the dynamics studied by V. I. Arnold, which exhibits what is known as Arnold's tongues. See this link.
  • 48.5k
21 votes

Rearrangements that never change the value of a sum

For the purpose of recording an answer rather than just a pile of links: Michael Hardy requires that if either limit exists then so does the other and in that case then they are equal? Let's call ...
18 votes
Accepted

How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{H_{n}^3}{(n+1)2^n} $?

By multiplying out the factor $H_n^3$, it is not too hard to see that your sum can be written as a rational linear combinations of special values of weight $4$ multiple polylogarithms. The Maple ...
  • 8,861
17 votes

Why do we teach calculus students the derivative as a limit?

Since this is community wiki, I'll feel free to share a possibly relevant anecdote; feel free to delete if you don't think this is an answer. I once had a freshman calculus student ask me if they'd ...
17 votes
Accepted

What is the structure associated to almost-everywhere convergence?

Yes, this defines a "convergence vector space". In fact, it's probably the original motivating example for the generalization. In Fréchet's thesis he discussed L-spaces, which are essentially ...
15 votes
Accepted

Factorial-based constant

I don't know about a name, but it does have a history. Knopfmacher, Odlyzko, Pittel, Richmond, Stark [D., not H.], Szekeres, and Wormald, The asymptotic number of set partitions with unequal block ...
15 votes

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I have no idea how to approach the general problem, but here is a quick observation: A. Let $\alpha = 2\beta$ so that $\beta$ is irrational if and only if $\alpha$ is so. Define $f$ by $f(x) = \log|2\...
13 votes

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This is really a comment on Joel's answer, but apparently too long. Let P be the forcing which adds a permutation of $\mathbb{N}$ by finite pieces (so $P$ is forcing-equivalent to Cohen forcing). ...
  • 2,440
13 votes
Accepted

Speed of convergence of $\zeta(2k)\to 1$?

Here is an explicit bound. The sum $\sum_{n > N} n^{-s}$ for real $s > 1$ is bounded by the integral $$\int_N^\infty x^{-s} = N^{1-s} / (s-1).$$ Therefore for any $N$ you have $$0 < \zeta(s) -...
13 votes

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

This is to prove the conjecture \begin{equation*} x_n\sim\sqrt3\,n^{1/2} \tag{1}\label{1} \end{equation*} (as $n\to\infty$). (For all integers $n\ge1$,) we have \begin{equation*} h_n:=x_{n+1}-...
12 votes
Accepted

Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

See the paper "Le poisson n'a pas d'arêtes" by Thierry Bousch. This is a joke that was explained to me much later. The set is considered to resemble a fish. The French word arête means both bone and ...
  • 21.9k
12 votes
Accepted

Does anyone recognize these polynomials? Need to compute a riemann lebesgue type limit

Maple says $$\sum _{j=0}^{n}{\frac {{n\choose j} \left( -z \right) ^{j}}{j!}}={L}_n \left(z \right)$$ Laguerre polynomials. See https://en.wikipedia.org/wiki/Laguerre_polynomials and go down to "...
  • 38.5k
12 votes
Accepted

Rate of convergence of $\frac{1}{\sqrt{n\ln n}}(\sum_{k=1}^n 1/\sqrt{X_k}-2n)$, $X_i$ i.i.d. uniform on $[0,1]$?

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\...
12 votes
Accepted

How bad can pointwise convergence in $C$ be?

This is just a standard application of the Baire category theorem. Proposition: Suppose that $X$ is a topological space where the Baire category theorem holds and $g_{n}:X\rightarrow[0,\infty]$ for ...
12 votes
Accepted

A limit problem

This equality is not true. E.g., if $f(u)=u$, $x=0$, and $a=1$, then the left-hand side of the equality is $1$, whereas its right-hand side is $2$. The OP has now switched the meaning of $r$ from ...
12 votes
Accepted

"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

The OP asks where this recursion relation might appear in a research context. It appears as a discretization of the Emden–Fowler nonlinear differential equation, $$f''(t)=t^{p}[f(t)]^q,$$ for $p=1$, $...
11 votes
Accepted

When does this interesting sum diverge?

In short, $$ \begin{cases} \text{when }1\leq x & \text{series diverges when }y\le1\\ \text{when }\frac{1}{2}<x<1 & \text{series diverges when }y\leq\frac{x}{2x-1}\\ \text{when }0<x\...
  • 10.7k
11 votes
Accepted

Infinite limit of ratio of nth degree polynomials

Here is an explicit formula for your ratio $r_n=\frac{n_n}{d_n}$: $$r_n= \frac{\sum_{k=0}^n\binom{n+k}{2k}(-x)^k} {\sum_{k=0}^n\binom{n+k+1}{2k+1}(-x)^k}.$$ Let $P_n(x)$ and $Q_n(x)$ be the numerator ...
11 votes

Ideal characterization of almost convergence

Such an ideal does not exist. Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So, $$\...
11 votes
Accepted

What does the abbreviation "p.p." mean in the context of convergence

This appears to be an abbreviation for presque partout, meaning almost everywhere. In the article you cite, reference is made to a paper of Hunt; the MathSciNet review for Hunt's paper (MR0236019) is ...
11 votes
Accepted

Subtracting the weak limit reduces the norm in the limit

The property you indicate is known as (strict) Opial’s Property (see https://en.m.wikipedia.org/wiki/Opial_property). It fails generally in reflexive spaces; in fact, it fails generally even for ...
  • 1,405
11 votes
Accepted

Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$?

We will compute the generating function, and use the method described in section 2 of this paper. Let $F_{m,n}=F(m,n)$. Consider the generating function $$G(x,y)=\sum_{m=0}^\infty\sum_{n=0}^\infty F_{...
10 votes
Accepted

A term for sequences whose mean is defined?

The standard term is Cesàro summable, named after Ernesto Cesàro. Note that a convergent sequence is also Cesàro summable (with the same limit), but the converse does not always hold. Edit. I ...
  • 29.2k
10 votes

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

Write it as $a_n(\alpha)$ to emphasize the dependence on $\alpha$. For any $\epsilon > 0$, $U(n,\epsilon) = \{\alpha: |a_n(\alpha)| < \epsilon\}$ is an open set containing $k/m$ for any integers ...

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