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 ...
- 1,555
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 ...
- 351
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
$$\lim_{...
- 172k
25
votes
Accepted
Convergence of Fourier series
Convergence in $L^p$, $p>1$.
True, by M. Riesz's Theorem. This is a standard topic in every harmonic analysis course, with several readable proofs.
Convergence pointwise almost everywhere, $p>...
- 824
25
votes
Accepted
Fraction of $S_n$ reachable by using every transposition once as $n\to\infty$?
The value is $1/2$.
The problem can be reformulated as follows: How to sort a non-sorted list $(a_1,...,a_n)$ that is a permutation of $\{1 .. .n\}$ with the parity the same as $n \choose 2$ by ...
- 9,272
24
votes
Accepted
Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
The product does not tend to the limit zero. For any irrational number $\alpha$ one can show that
$$
\limsup_{N\to \infty} \prod_{n=1}^{N} |1- e(n\alpha)| = \infty. \tag{1}
$$
(Here I use the usual ...
- 43.2k
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.
- 50.5k
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 ...
- 8,911
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 ...
- 3,202
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=\...
- 20.2k
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 ...
- 38.4k
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\...
- 564
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$ ...
- 30.9k
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) -...
- 8,791
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}-...
- 110k
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 ...
- 22.4k
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 "...
- 40k
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}{\...
- 110k
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 ...
- 27.3k
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 ...
- 110k
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 $p=1$, $...
- 172k
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 $...
- 4,913
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 ...
- 76.6k
11
votes
Accepted
When does this interesting sum diverge?
In short,
$$ \begin{cases}
\text{when }1\leq x & \text{series diverges when }y\le1\\
\text{when }\frac{1}{2}<x<1 & \text{series diverges when }y\leq\frac{x}{2x-1}\\
\text{when }0<x\...
- 11.2k
11
votes
Accepted
Infinite limit of ratio of nth degree polynomials
Here is an explicit formula for your ratio $r_n=\frac{n_n}{d_n}$:
$$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}.$$
Let $P_n(x)$ and $Q_n(x)$ be the numerator ...
- 41.2k
11
votes
Ideal characterization of almost convergence
Such an ideal does not exist.
Indeed, suppose the contrary, and let $I$ be such an ideal. The sequence $(x_n)=(1,0,1,0,\dots)$ is almost convergent, and therefore $I$-convergent, to $1/2$. So,
$$\...
- 110k
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 ...
- 4,155
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 ...
- 1,425
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_{...
- 2,464
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 ...
- 8,352
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