Convergence of sequences and different modes of convergence.

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1answer
27 views

Computing $\mathrm{erfi}(\theta)\exp(-\theta^2)$:

I'm looking to compute $f(\theta):= \mathrm{erfi}(\theta)\exp(-\theta^2)$ as efficiently as possible, to double precision, with a fairly wide radius of converge. Computing $\mathrm{erfi}(\theta)$ ...
0
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3answers
43 views

Does the following matrixpowers converge?

Given a $3\times 3$ Matrix: $$A= \begin{bmatrix} -2 & -16.8 & -16.8& \\ -1 & -4.8 & -5.8 \\ 1.4 & 8 & 9\\ \end{bmatrix} $$ Determine whether the powers $A^n, ...
2
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0answers
22 views

Convergence of sum of exp. decaying pdf // When does L^2 convergence imply a.s. convergence?

The problem: Let $X_t^i$ $(i \in Z)$ be integer valued random variables on the same probability space. Let $m: Z \rightarrow R $ be a symmetric probability density on the integers which has ...
0
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0answers
24 views

Exponential convergence of controlled variables

I am reading a paper and I don't understand why, after some math they say that the controlled variables $$ \dot{\psi}_{13} $$ and $$ \dot{\psi}_{23} $$ converge exponentially. This is the paragraph ...
3
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1answer
66 views

Limit of $nx_n$

Okay, so I have this problem: Given the sequence $(x_n)_{n\in\mathbb{N}}$ defined by $x_{n+1}=\dfrac{3x_n^2}{(1+x_n)^3-1}$, with $x_1>0$, find $\displaystyle\lim_{n\to\infty}x_n$ and ...
6
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2answers
211 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
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0answers
13 views

Does the magnus convergence test not hold for the factorization of second order differential operators?

Given the operator \begin{align} H = V(x)-\partial_x^2 \end{align} and given an eigenfunction $\phi_0(x)$ such that $H\phi_0=0$ with a zero eigenvalue, I can factor $H$ into \begin{align} H = h_+h_- ...
3
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2answers
64 views

How to prove $( \sum_{n=1}^{\infty} |x_n|^2)^{1/2} \le \sum_{n=1}^{\infty} |x_n|$ (cauchy-product)

I am having this: $ (\ x_n)\ _{n \in \mathbb N} $ is sequence in $\mathbb C$, so the series $\sum_{n=1}^{\infty} |x_n|$ converges. I've already proved that the series $\sum_{n=1}^{\infty} |x_n|^2$ ...
0
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2answers
23 views

Determine completeness of a metric space

Let $X := (0,\infty)$ and $\left(X, \rho: X\times X\rightarrow (0,\infty), (x,y)\mapsto\rho(x, y):=\left|\frac{1}{x}-\frac{1}{y}\right|\right)$ a metric space. Determine whether it is complete or ...
1
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1answer
21 views

Convergence in distribution with exponential limit distribution

Let $X_1,X_2, \ldots$ be independent, identically distributed, positive random variables with probability density function $f$, which is continuous in $(0, \infty)$ and $\lambda :=\lim_{x \searrow 0} ...
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3answers
46 views

My proof of uniqueness of limit (of sequence)

Should I try to write a direct (i.e. non$-$by-contradiction) proof instead of the below proof? (I was told that mathematicians prefer direct proofs.) We consider a convergent sequence which we ...
1
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1answer
33 views

Catalan divided by increasing powers

Find the value of the converging sum $\frac{1}{k}+\frac{1}{k^2}+\frac{2}{k^3}+\frac{5}{k^4}+\frac{14}{k^5}+\frac{42}{k^6}+\frac{132}{k^7}...$, where the numerators are the catalan numbers in ...
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0answers
7 views

Increase rate of convergence for steepest descent

How can one find a transformation matrix $T$ for $y=Tx$ that decreases the condition number of the Hessian of a quadratic function and decreases the iteration time?
2
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1answer
40 views

series convergence. $\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt[3]{n}+ (-1)^{\frac{n(n+1)}{2}}}$ [duplicate]

$$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt[3]{n}+ (-1)^{\frac{n(n+1)}{2}}}$$ None of convergence tests I know (Leibnitz, Dirichlet, Abel) works becaues of the denominator. I know that sum of convergent ...
1
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0answers
23 views

Prove convergence of $(1-\frac xk)^k$ as $k\to\infty$ using arithmetic-geometric mean

Define $f(x):=x^{t-1}e^{-x}$. For $k=1,2,\dots$ let $$f_k(x)=\begin{cases}x^{t-1}\left(1-\frac xk\right)^k & 0<x<k\\0&k\le x\le \infty\end{cases}$$ Show that $f_k(x)\to f(x)$ and ...
1
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1answer
25 views

series convergence. Dirichlet test

$$\sum(-1)^{\lfloor\frac{n^3+n+1}{3n^2-1}\rfloor}\frac{\ln(n)}{n}$$ I thought about using Dirichlet test. $\ln(n)/n$ is a decreasing sequence that tends to 0 but I have problem with proving it. I also ...
3
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1answer
38 views

If $\int _0^1|f_n(t)-f(t)|dt \rightarrow 0 $ then $\lim\limits_{n\rightarrow \infty}\int _0^1f_n(t)dt =\int _0^1f(t)dt$.

If $\int _0^1|f_n(t)-f(t)|dt \rightarrow 0 $ as $n\rightarrow \infty$ then how to prove that $$\lim\limits_{n\rightarrow \infty}\int _0^1f_n(t)dt =\int _0^1f(t)dt$$ Any hints?
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1answer
35 views

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: ...
7
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1answer
89 views

My proof of $\lim_{n\to\infty} (1 / n) = 0$

Is my proof correct? We consider the sequence \begin{equation*} (x_n)_{n = 1}^{\infty}, \qquad \text{where} \qquad x_n = \frac{1}{n}. \end{equation*} $\textbf{Theorem.}$ ...
-1
votes
1answer
68 views

Given $a_n \to 0$, weakest condition on $b_n$ such that $a_n b_n \to 0$ [closed]

Let $a_n$ be "some" sequence such that $a_n \to 0$. I want to know the weakest condition on $b_n$ which will make $a_nb_n \to 0$. If $b_n$ is bounded then the claim is true. Can I relax this ...
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0answers
33 views

Semigroups & Generators: Entire Elements: Construction

Problem Given a Banach space $E$. Consider a $\mathcal{C}_0$-group(!): $T:\mathbb{R}\to\mathcal{B}(E)$. Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)$$ (The domain being those ...
0
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2answers
112 views

Showing a function converges to e

I'm trying to show that the following limit converges to $e$: $$\lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}} = e$$ where $e$ is defined as follows: $$ e = \lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$$ I ...
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0answers
27 views

Is my proof of a dominated convergence corollary using Egorov's theorem right?

Theorem: If $\mu(\Omega) < \infty$ and the $f_n$ are uniformly bounded, then $f_n \to f$ almost everywhere implies $\int f_n d\mu \to \int fd\mu$. This is a simple consequence of Lebesgue ...
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2answers
40 views

Convergence on every compact set implies convergence almost everywhere

Suppose I have a sequence of functions {$u_n$} that converges to $v$ uniformly on every compact subset of $\mathbb{R}^n$. Suppose further that {$u_n$} converges to $u$ in $L^1$ for every compact ...
2
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3answers
61 views

Is there any way to get convergent series if I know the sum?

If I am given the limit of a convergent series is there any way that I can find the series? Is it possible that for any given limit there are infinitely many or no solutions at all?
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0answers
31 views

the continuity of argmin

how can I show the minimum of a converging sequence of continuous functions is converging to the minimum of the limit of that sequence.i.e. $$\lim_i \text{argmin}_{x} f_{i}(x)=\text{argmin}_{x} \lim_i ...
4
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2answers
65 views

How to calculate what this power series converges against? (double factorials)

I'm working on my physics master course homework and I'm given the following equation out of nowhere: $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over ...
0
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2answers
34 views

Prove/disprove that $\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$ given $|x|<1$

For $|x|<1$, if the following statement is true, how to prove it? If not, how to disprove it? $$\sum_0^\infty a_nx^n = 0 \rightarrow a_n = 0 \text{ for all }n$$ In case $x$ takes any real value, ...
0
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1answer
34 views

Determine all $a\in\mathbb{R}$ so that a series converges

How do I determine all $a\in\mathbb{R}$ for a series $\sum \limits_{n=1}^\infty (-1)^n \cdot \frac{a^n}{n}$ so that the series converges? I know that the series converges for $a=1$ And I ...
0
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1answer
26 views

General term of a series that subtracts the square root of every square.

I'm trying to figure out the general term for a series where you subtract from every perfect square number, its square root. So ...
1
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2answers
50 views

Investigate convergence of the series

Investigate convergence of the series: $$\left( \frac{n^2+3n+10}{n^2+5n+17} \right)^{n^2 (\sqrt{n+1}-\sqrt{n-1})}$$ It should be solvable with simple manipulations with the formula, i guess, but how ...
2
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1answer
43 views

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
1
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1answer
48 views

Mollifiers: Integral Convergence

Why do these integrals converge: $$\varphi\in\mathcal{C}_0^\infty:\quad\frac{1}{\tau}\int_0^\tau\varphi(s)\mathrm{d}s\to\varphi(0)\quad(\tau\geq0)$$ I tried to figure it out via substituting: ...
1
vote
1answer
21 views

Measurability of an a.e. pointwise limit of measurable functions.

Suppose that $(f_n)_n$ is a sequence of measurable functions on a set $E$ and that $f_n \to f$ a.e.on $E$. Does this imply that $f$ is measurable? I know that pointwise limit of measurable function ...
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0answers
14 views

trouble with convergent and divergent series

Determine whether zn=nth root of(e^n^2(i-1)) is convergent or divergent? i have having trouble with this. How to proceed with this?
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1answer
40 views

Study the convergence of the sequence of functions $f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$ (convergence in measure, pointwise and in $ L^2(R ^d)$

Study the convergence of the sequence of functions $$f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$$ (convergence in measure, pointwise and in $ L^2(\mathbb{R} ^d)$). Let f be a measurable function such ...
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0answers
17 views

Proof of Kuratowski-Wojdyslawski theorem

I was reading the Wikipedia page on Kuratowski Embedding, and the following result is stated: The Kuratowski–Wojdysławski theorem states that every bounded metric space $X$ is isometric to a ...
2
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1answer
37 views

Bounding the difference between $H_N$ and $\log N$

we consider the relation given: $$\int_n^{n+1}\frac{1}{x}dx < \frac{1}{n} < \int_n^{n+1}\frac{1}{x-1}dx$$ For $n>1$. We are to show ...
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32 views

Apply the definition of a limit of a function to determine $ \lim_{x\rightarrow 0} x^{a} $ where $ a \in \mathbb{R} $ is positive

As stated above. I know how to verify the existence of a limit, but have no idea how to find the limit, any ideas? Is this sufficient? Using the definition of convergence when $x = 0 $ we get $ ...
2
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1answer
107 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta ...
1
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1answer
72 views

Convergence of multiple zeta function

The following term:$$\zeta(k_1,k_2,...,k_n)=\sum_{m_1>m_2>\cdots>m_n>0}\frac{1}{m_1^{k_1}m_2^{k_2}\cdots m_n^{k_n}}, m_i\in\mathbb{N}, k_i\in\mathbb{N}$$ is called the "multiple zeta ...
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0answers
14 views

Convergence of Distribution Functions

This is paragraph from de Haan's Extreme Value Theory (2006, p4). Let $F$ be a cumulative distribution function, $a_n$ a sequence of positive constants and $b_n$ a sequence of real numbers. Suppose ...
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1answer
11 views

Is linear convergence norm invariant?

Let $\|\cdot\|_a,\|\cdot\|_b$ be two norms on $\Bbb R^n$ and $(x^k)_{k\in\Bbb N}\subset \Bbb R^n$ a sequence such that there exists $0<\alpha <1$ with $ \|x^{k+1}\|_a \leq \alpha \|x^k\|_a$ for ...
0
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1answer
45 views

Proving that a recursive sequence converges

The sequence is defined as $ x_{n} = \sum_{k=1}^{n} \frac{1}{3^{k}} $ I have re-written the sequence like so: $x_{1} = \frac{1}{3} $ and $ x_{n} = \frac{1}{3^{n}} + x_{n-1}$ Now it's easier to ...
1
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1answer
40 views

Weak/strong law of large numbers for dependent variables with bounded covariance

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, ...
0
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1answer
24 views

Show this converges in distribution to 0

Let $\{ X_n:n \geq 1 \}$ such that $$f_{X_n} = \begin{cases} (n-1)/2 &\mbox{if } -1/n <x<1/n \\ 1/n & \mbox{if } n<x<n+1 \end{cases}$$ Show that this converges to $0$ in ...
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0answers
45 views

Difference between convergence in measure and almost everywhere convergence

We say$$f_n \rightarrow f$$ almost everywhere on $\Omega$ $iff$ there exist N in sigma algebra $F$ such that $\mu$(N) = $0$ and $$f_n(\omega)\rightarrow f(\omega)$$ for all $\omega$ in $N^c$ and ...
4
votes
3answers
537 views

Why is the convergence absolute?

There is one thing my book uses in a proof after Abels theorem which I do not understand: Lets say that $\Sigma_{n=0}^\infty a_n$ converges. For $0\le x<1$, we look at $\Sigma_{n=0}^\infty a_n ...
1
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1answer
18 views

Convergences of square of difference in probability implies convergence in probability

Consider real valued random variables $X,X_n,n\in\mathbb{N}$. If $(X_n-X)^2\xrightarrow{P} 0$ then $X_n\xrightarrow{P} X$. I tried using Chebyshevs inequality (which seems to be the usual ...
3
votes
3answers
174 views

Convergence series of positive numbers

Let $x_1,x_2,...,x_n,...$ be positive numbers. If $\sum _{n=1}^\infty x_n$ converges, how do I show that $$\sum_{n=1}^\infty \frac{x_1+2x_2+3x_3+...+nx_n}{n(n+1)}$$ is also convergent?