Convergence of sequences and different modes of convergence.

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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?
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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 ...
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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 ...
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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 ...
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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: ...
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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*} $\textbf{Theorem.}$ ...
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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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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 ...
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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 ...
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3answers
60 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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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: $$ ...
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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: ...
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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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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
39 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 ...
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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 $ ...
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1answer
106 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 ...
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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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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 ...
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1answer
43 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 ...
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1answer
39 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, ...
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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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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 ...
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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 ...
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1answer
17 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 ...
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173 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?
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40 views

Linear functional norm

We are given $T:l_2\rightarrow \mathbb{R}$ such that $T({x_n})=\sum_{n=1}^\infty \frac{1}{\sqrt{n}}x_{3n}-\sum_{n=1}^\infty\frac{1}{n}x_{2n}$. I would like to know what the norm of that functional is ...
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1answer
39 views

Prove convergence in distribution

I need help with the following problem. Let $X_{n1}, X_{n2}, . . . , X_{nn}$ be independent random variables, with the same distribution as follows. Let for k = 1, 2, . . . , n och n = 1, 2, . . . , ...
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1answer
41 views

Convergence of series in a Hilbert Space

I'm hoping for some help on the following question. I haven't gotten very far: Let $\{h_n\}_{n\geq 1}$ be a sequence of vectors in a Hilbert space $H$ with the property that $(h_n-h_m)\perp h_m$ ...
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0answers
56 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
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1answer
33 views

How to choose a convergence test for a given infinite series?

I have a general question when it comes to deciding if an infinite series is convergent or divergent. The tests im familiar with are ; Ratio test, Direct comparison test, Limit comparison test, Root ...
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1answer
31 views

Convergence of series for specific values of $\lambda$.

Let $\lambda$ be a positive real number. For which values of $\lambda$ does the following series converge? $$\sum_{n=1}^\infty \frac{n^{-\lambda}}{1+\lambda^{-n}}$$ I can see that the series ...
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1answer
36 views

Value of given series $\sum_{n=1}^{n=\infty}\frac{\cos{\frac{n\pi}{3}}}{n+1}$ and $\sum_{n=1}^{n=\infty}\frac{\sin{\frac{n\pi}{3}}}{n+1}$

I was doing a problem,after some calculation it all came down to the value of series $$\sum_{n=1}^{n=\infty}\frac{\cos{\frac{n\pi}{3}}}{n+1}$$ and $$ ...
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1answer
68 views

Evaluate $\int_{-\infty}^{\infty} \frac{\log(1+x^2) dx}{1+x^2}$ Using Complex Analysis

Evaluate: $$\int_{-\infty}^{\infty} \frac{\log(1+x^2) dx}{1+x^2}$$ Using complex analysis, contour integration. This function has no poles at all. Try the contour $C$ Obviously, ...
4
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1answer
61 views

Series convergence - factorial over products

Does $$ \sum_{n\ =\ 1}{n!\over \left(\,\sqrt{\,2\,}\, + 1\,\right) \left(\,\sqrt{\,2\,}\, + 2\,\right)\ldots \left(\,\sqrt{\,2\,}\, + n - 1\,\right)\left(\,\sqrt{\,2\,}\, + n\,\right)}\quad $$ ...
4
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5answers
272 views

Does ${\sum_{n=1}} \log(1+{1\over n})$ diverge or converge?

How do I find out if ${\sum_{n=1}} \log(1+{1\over n})$ diverges or converges? Wolfram recommends me to use comparsion test, but I do not know series which diverges and less than this.
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2answers
41 views

Dominated convergence theorem on $e^{ix}$

I am considering first: $$\lim_{n \to 0} \int_{0}^{\pi} e^{ine^{ix}} dx$$ To bring the limit inside I need to apply the dominated convergence theorem. Keep in mind I have no knowledge of measure ...
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1answer
34 views

Show that this sum is divergent

How can I argue that this sum i divergent? $$\sum_{j=1}^\infty \left( \frac{1}{j\beta}+\frac{\delta}{(j-1)\beta^2}+ \frac{\delta^2}{(j-2)\beta^3}+ \cdots + \frac{\delta^{j-1} }{\beta^j} \right)$$ ...