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

**-1**

votes

**0**answers

18 views

### Limits of an indeterminate form $\lim_{t\to\infty} (a+b(-m)^t)/(c+d(-m)^{t-1})$ [on hold]

I'm trying to solve the limit of the following indeterminate form:
$$\lim_{t\to\infty} \frac{a+b(-m)^t}{c+d(-m)^{t-1}}$$
where $t=1, 2, 3, \cdots$ denotes time and all the coefficients are positive ...

**0**

votes

**0**answers

15 views

### Are there examples of functions with Nesterov's convergence bound between convex quadratic and strongly convex cases?

Are there examples of simple and strongly convex functions for which the convergence bound of Nesterov’s Accelerated Gradient Method is better than Nesterov’s bound for strongly convex case ($\sqrt{1 -...

**0**

votes

**1**answer

33 views

### Convergence of an inhomogeneous markov chain

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial ...

**0**

votes

**0**answers

12 views

### Conditions for convergence to non-isolated fixed points

Consider a dynamical system of the form
$$
\dot{x}=f(x), \quad x\in X,
$$
and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...

**0**

votes

**0**answers

54 views

### The necessary and sufficient condition for exchanging the integration process and the limit process [closed]

We only consider real functions here. It can be shown that
\begin{eqnarray*}
\int_{0}^{\infty}dx\frac{\sin x}{\sqrt{x}} & = & \int_{0}^{\infty}dx\frac{\cos x}{\sqrt{x}}=\sqrt{\frac{\pi}{2}}
\...

**2**

votes

**1**answer

45 views

### Cubic splines convergence?

I am looking for a basic, classical, result on approximating a smooth function using cubic and linear splines. Is there a reference on the convergence, in some sense, of the splines to the function of ...

**1**

vote

**0**answers

46 views

### The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows
\begin{equation}\label{cp26}
F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...

**1**

vote

**0**answers

66 views

### Law of large numbers for random functions?

Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n =...

**0**

votes

**0**answers

51 views

### Heat kernel with unbounded potential function

If we consider the Laplace/Schrodinger operator $H=−Δ+V$ on $R^n$, where $V$ is positive and goes to $\infty$ as $|x|\rightarrow \infty$. Then we can prove that $\exp{-tH}$ is trace class for any $t&...

**0**

votes

**1**answer

29 views

### Limits of argmin ratios and sums

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms
\begin{equation}
\lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\...

**14**

votes

**2**answers

313 views

### Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...

**1**

vote

**0**answers

58 views

### Converge of measures [closed]

Good night, we have the following:
Let $(X,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...

**3**

votes

**0**answers

108 views

### Limits of definable maps

For sequences of semialgebraic maps there is the following result:
Let $(f_{n}: ]0,1[^d \to ]0,1[)_{n \in \mathbb{N}}$ be a sequence of continuous semialgebraic maps of bounded degree such that $(...

**2**

votes

**0**answers

132 views

### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

**0**

votes

**1**answer

73 views

### Heights of multiples of rational points on elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the ...

**13**

votes

**1**answer

183 views

### Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...

**3**

votes

**1**answer

190 views

### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...

**1**

vote

**1**answer

63 views

### Inequality implies locally uniform convergence of a series

We have the inequality
$$\alpha_n(t) \le 4\pi(3\pi)^{1/3} \exp\left\{\int_0^t(1+3F_P(\sigma)) \, d\sigma \right\} \cdot \int_0^t P(\sigma)^2(3CD_1^2)^{1/3}\alpha_{n-1}(\sigma) \, d\sigma$$
for $n=2,3,\...

**0**

votes

**1**answer

101 views

### Is the set of Cauchy spaces a lattice? [closed]

Is the set of all Cauchy spaces (ordered by set-theoretic inclusion) on some (fixed) set:
a join-semilattice?
a meet-semilattice?
a complete lattice?

**3**

votes

**1**answer

101 views

### π based on the perimeter of inscribed polygons [closed]

So, last year I got obsessed with the idea of finding a way to calculate π that wasn't already done. After reading some history, the Greek idea of measuring polygons inscribed within circles and ...

**0**

votes

**0**answers

38 views

### On the continuity of Riemann-Liouville integral

For a function $f:(0,1)\to\mathbb{R}$, the Riemann-Liouville integral of $f$ is defined by
$$
(I^{1-\nu}f)(t):=\frac{1}{\Gamma(1-\nu)}\int_{0}^{t}\frac{f(s)}{(t-s)^{\nu}}ds,
$$
where $\nu\in(0,1)$ is ...

**-4**

votes

**1**answer

148 views

### Does $\sum_n \frac{\sin n}n$ converge absolutely? [closed]

Using Dirichlet's test, one can prove that $\sum_{n\geq 1} \frac{\sin n}n$ converges. Does it converge absolutely?

**7**

votes

**2**answers

394 views

### Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...

**4**

votes

**2**answers

176 views

### A particular Diophantine approximation of $\pi/2$

I have asked this question in math.stackexchange without any answer, so I have decided to post it here too.
Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$
...

**1**

vote

**0**answers

86 views

### “Nice” limits of sequences of smooth embeddings

Consider smooth embeddings of a manifold $M$ into some $\mathbb{R}^n$. If a sequence $f_k : M \to \mathbb{R}^n$ of such embeddings converges to some continuous function $f : M \to \mathbb{R}^n$, then ...

**20**

votes

**3**answers

524 views

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

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...

**8**

votes

**1**answer

286 views

### Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...

**2**

votes

**0**answers

135 views

### On a matrix algorithm involving rank-one projections

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
X_{k+1}=\frac{1}{N}\sum_{i=1}^...

**19**

votes

**1**answer

317 views

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

I made a throwaway comment on math stackexchange the other day that got me thinking about the following question. Let
$$ f_n (\alpha) = \frac1n \sum_{k=0}^{n-1} e(2^k\alpha),$$
where $e(x) = \exp(i2\...

**3**

votes

**1**answer

88 views

### How to show monotonocity and the limit? [closed]

Let me reformulate my recent question.
Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density:
$$\phi(x) = C\left\{ \begin{array}{lcc}
\sqrt{...

**3**

votes

**1**answer

277 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 $$I=\sum_{...

**5**

votes

**2**answers

188 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 $\mathbb{Q}...

**0**

votes

**1**answer

62 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

168 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 \binom{z}{n}...

**2**

votes

**2**answers

166 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 $f(x)=\...

**4**

votes

**1**answer

634 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

192 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 \left\{\frac{...

**5**

votes

**1**answer

150 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

119 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

355 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+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\...

**1**

vote

**0**answers

106 views

### Strong convergence

From this paper: https://www.researchgate.net/profile/Claudianor_Alves/publication/264957577_Existence_and_multiplicity_of_solution_for_a_class_of_quasilinear_equations/links/54d5e9a00cf25013d02c1e25....

**7**

votes

**0**answers

378 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

322 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

110 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

87 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

39 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

92 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

62 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

104 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

189 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:
$$
\begin{...