Convergence of series, sequences and functions and different modes of convergence.

**-2**

votes

**0**answers

47 views

### Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated.
I got struck ...

**-3**

votes

**0**answers

18 views

### Square wave in the limit of infinite frequency [on hold]

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity?
Does it depend on how the function is written down (e.g. defined as ...

**3**

votes

**1**answer

235 views

### Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...

**5**

votes

**2**answers

153 views

### When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of ...

**0**

votes

**1**answer

54 views

### Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$:
$\require{AMScd}$
\begin{CD}
a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...

**2**

votes

**1**answer

144 views

### Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$

Is the following limit result correct: $$\lim\limits_{x \to 0^{+}}\sum\limits_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle ...

**1**

vote

**2**answers

110 views

### Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set ...

**4**

votes

**1**answer

612 views

### When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...

**1**

vote

**1**answer

172 views

### Convergence of a double sum involving prime numbers

This has been moved from math.stackexchange;
I am attempting to prove/disprove convergence of the following sum
$$ \lim_{n \to \infty} \frac{1}{n} \sum_{p \leq n} \sum_{k=0}^\infty \ln p ...

**5**

votes

**1**answer

134 views

### Rate of Convergence of Borwein Algorithm for computing Pi

In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't ...

**3**

votes

**0**answers

89 views

### Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy ...

**6**

votes

**1**answer

343 views

### Factorial-based constant

Am looking for a name for:
$$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$
$$=2.529477472079152648180116154253954242$$
Wolfram|Alpha
Expanding the formula gives:
...

**1**

vote

**0**answers

103 views

**7**

votes

**0**answers

341 views

### Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7:
Lemma: Let $K$ be a ...

**2**

votes

**2**answers

315 views

### Is $\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$ finite for every $k$?

I would like to check if this limit :$$\liminf \frac{\sigma_{k}(({2}^{m-1})({2^m-1}))}{\phi_{k}(({2}^{m-1})({2^m-1}))}$$ finite for every $k$?
where :$\phi_{k}$ is iterating Euler - totient function ...

**4**

votes

**0**answers

81 views

### On contractive properties of a nonlinear matrix algorithm

I’m stuck in a problem that concerns a nonlinear iterative matrix algorithm.
Although the problem is quite complicated to explain I’ll try to describe it in a simple way, neglecting unnecessary ...

**0**

votes

**1**answer

68 views

### Almost sure convergence of smallest eigenvector of diagonal matrix

I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...

**1**

vote

**0**answers

35 views

### convergence of unconstrained convex optimization

I encounter an optimization problem. The simplified version is like following:
Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. ...

**2**

votes

**1**answer

77 views

### Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...

**2**

votes

**0**answers

56 views

### Proving convergence is impossible for a sum of hyperbolic cosines

Suppose that $z$ is some complex value. Is it possible to prove that
$$\lim_{n \rightarrow \infty} \sum_{j = 1}^n {\sqrt{n \over j}} \cdot \cosh(z \log {n \over j}-\operatorname{ Arccoth} (2z))
$$
...

**1**

vote

**1**answer

96 views

### Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each ...

**3**

votes

**2**answers

162 views

### Uniform Convergence of Moment Generating Function

In the article, "The Empirical Moment Generating Function" by Csörgö, the author defines the empirical moment generating function for a sample of $n$ variables $X_1,X_2, \dots, X_n$ as:
$$
...

**6**

votes

**2**answers

258 views

### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that:
$$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$
with $p_n$ ...

**3**

votes

**2**answers

149 views

### Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
...

**5**

votes

**1**answer

260 views

### The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$
I am not really sure quite where to start here as I am ...

**0**

votes

**0**answers

28 views

### Convergence of generalized inverses

During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation to post more links), I've got stuck, trying to understand more deeply one of ...

**3**

votes

**1**answer

76 views

### Uniform convergence of 2-norm of a multinomial vector

Let $(X_1,X_2,\ldots,X_k)$ be distributed according to a multinomial distribution with parameters $(n;p_1,p_2,\ldots, p_k),$ i.e.
$$P(X_1=n_1,\ldots,X_k=n_k) = {n\choose n_1,n_2,\ldots,n_k} ...

**2**

votes

**0**answers

98 views

### Notion of convergence on a dense subset

My motivation for this question is as follows.
Consider the set $D$ of cadlag functions on $(0,1)$ and its subset $D^\uparrow$ of non-decreasing cadlag functions.
Each $f \in D$ has at most countably ...

**1**

vote

**1**answer

150 views

### limit and combinatorics

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit:
$\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$
When ...

**3**

votes

**1**answer

106 views

### Weak convergence of random variables in $L^2$ and vague convergence

Dumb question: Let $X_n:\Omega \to \mathbf{R}$ be a sequence of $L^2(\Omega,\Sigma,\mathbf{P})$ random variables that has a weak limit $X$ in $L^2$.
Suppose also that $\mu_n$, the distributions of ...

**2**

votes

**0**answers

180 views

### Is this limit of a sequence of sets correct? [closed]

The following calculation of a limit of a sequence of sets according to https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior has been claimed to be wrong in MSE ...

**4**

votes

**1**answer

132 views

### Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...

**2**

votes

**1**answer

104 views

### Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...

**2**

votes

**0**answers

59 views

### Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in ...

**2**

votes

**2**answers

95 views

### Special filters in the algebra of regular open sets of a topological space

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is:
$$A\in\mathrm{r}\mathscr{O}\iff ...

**2**

votes

**1**answer

72 views

### Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\ ...

**2**

votes

**0**answers

112 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**25**

votes

**5**answers

964 views

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

This will not be altogether unrelated to this earlier question.
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...

**1**

vote

**1**answer

112 views

### Limit of largest eigenvalue [closed]

For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above?
In other words does the following limit exists:
...

**4**

votes

**0**answers

79 views

### Convergence of a particular double sum [closed]

Consider the following double sum:
$$Q(n)=\frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}\left [ \partial _{ij}lnf\left ( x \right ) \right ]^2$$
where $\partial_{ij}$ is the partial second order ...

**-1**

votes

**1**answer

243 views

### Are limits decidable? Should definitions be decidable? [closed]

This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition:
There cannot exist a Turing Machine $M$ which, given a ...

**18**

votes

**1**answer

462 views

### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...

**2**

votes

**0**answers

90 views

### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
...

**13**

votes

**2**answers

1k views

### Is there a $q$-L'Hospital's Rule?

Let $\binom{n}{j}_q$ be a $q$-binomial coefficient and $(x;q)_n = (1-x)(1-qx)\cdots(1-q^{n-1}x).$
Consider the sum $$f(n,m,r,k)= \sum\limits_{j = 0}^{2n} {( - 1)}^{ j}q^{mj^2+rj}
...

**2**

votes

**0**answers

59 views

### Decomposition of the space according to the Ergodic Theorem

Given a space $(X, T)$, it is well known that for every $T$-invariant ergodic measure $\mu$, there exists a set $E_\mu$ of $\mu$-measure $1$ s.t. for every "nice" function $f$
$$ ...

**0**

votes

**0**answers

82 views

### Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$:
$h(\eta_{n}) \rightarrow ...

**6**

votes

**3**answers

325 views

### Natural topologies for the space of rational functions

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree ...

**0**

votes

**0**answers

50 views

### Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)

Consider the following Volterra integral equation
$g(t)=∫_0^tK_n(t,s)w_n(s)ds$
where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...

**1**

vote

**0**answers

97 views

### Bound on $g(n+1)/g(n)$ for Landau's function

I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that
$\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$
(stated, but not proved in "On ...

**2**

votes

**1**answer

81 views

### Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...