45
votes

Accepted

### A challenging (for me) limit calculation

This limit converges to $\frac{\sqrt3}2$. The idea is that $\sin(x) = x - \frac{x^3}6 + O(x^5)$, so we start with $\frac1{\sqrt n}$ and repeatedly subtract $\frac{x^3}6$. We can approximate this ...

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_{...

25
votes

Accepted

### 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 ...

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 ...

24
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.

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 ...

18
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 ...

16
votes

### Solving a limit about sum of series

An alternative to Carlo Beenakker's argument, which explains the appearance of $\pi$ geometrically, is as follows: With $A(t)=\sum_{n=0}^{\infty}t^{n^2}$ we have
\begin{equation}
\frac{1}{1-t}A(t)^2=\...

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\...

15
votes

Accepted

### Approximate size of the image of functions $f:[n]\to[n]$

For each $i \in [n]$, the probability that $i$ is in the image of a random function $f: [n] \to [n]$ is $1 - \frac{(n-1)^n}{n^n}$. By linearity of expectation, the expected size of the image of $f$ ...

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 ...

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 "...

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 $p=1$, $...

12
votes

Accepted

### A limit involving the quotient of two sums

First of all, it seems like the value of the limit is more like $1.27...$, and not $2.27...$.
Using some heuristics outlined below it is possible to find the limit:
$$
a=1.278464542761...,
$$
where $...

12
votes

### Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped

The probability of not winning is
$$
\prod_{T=1}^\infty \left(1 - \frac1{2^T} \right)
= \frac12 \frac34 \frac78 \frac{15}{16} \cdots
= 0.28878809508660 \ldots ;
$$
that's a well-known constant (equal ...

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\...

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 ...

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_{...

11
votes

### Twice continuously differentiable implied by existence of limit

This is more of a long comment than answer. First, the analogous statement for the first derivative is already non-trivial, although not very difficult, see Aull, Charles E. "The first symmetric ...

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