Convergence of sequences and different modes of convergence.

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Square wave in the limit of infinite frequency

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 ...
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3answers
23 views

Uniform convergence with two limits

I'm doing a question investigating uniform convergence of a function and I need something cleared up if possible. $f_n(x) = \frac{x^n}{1+x^n}$ on the interval $[0,1]$. Now, pointwise, this turns ...
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2answers
64 views

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
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37 views

Series converges point-wise [on hold]

$$f_{n}=\sum_{n=1}^{\infty }\frac{x^{4}}{(1+x^{4})^{n}}$$ Show that it converges point-wise on $\mathbb{R}$, but not uniformly on $\mathbb{R}$. My attempt: I think, we should use Weierstrass's M ...
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1answer
25 views

Domain of $\lim_{n\to\infty}|(n+1)x| < 1$

In the process of doing the ratio test (for testing convergence of a function), I have the following issue. I am trying to find the convergence domain of $x$ in the following function: $$ ...
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1answer
23 views

Show the convergence of a sequence using a convergent sequence in a metric space

Let $X$ a metric space with metric $d$ and let $P \subset X$ not empty. Let $f :X \to \mathbb{R}$ a function and define $f(x) = \inf\{d(x,y): y \in P\}$. Let $(x_n)_{n \in \mathbb{N}}$ a convergent ...
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Bounded, monotone, convergent [on hold]

Study if the sequence $X_n$ is bounded, monotone and convergent. If the sequence is convergent, find also its limit. $X_1 \in (0, 1), X_{n+1}=2X_n +13,n \in N$
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1answer
47 views

convergence of $f(x+y)$ to $f(x)$ for $y >$ to $0$ in $L^1$.

I have to show the following: for $f \in L^1(S^1)$ Show that the map $f(x) \to f_y = f(x+y)$ is continuous in the distance of $L^1(S^1)$. i.e. $\lim_{y \to 0} ||f_y-f||_1 = 0$ I am supposed to use ...
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2answers
46 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
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1answer
58 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...
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1answer
28 views

use L1-convergence to show integral convergence

Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 ...
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18 views

Non-linear simultaneous recurrence system

Given a non-linear, non homogeneous, discrete time recurrence system: $a_i^t = f_i(a_1^{(t-1)},a_2^{(t-1)},\ldots,a_k^{(t-1)},C_1)$, for all $i\in [k]$ where $C_1,\ldots,C_k$ are constants and each ...
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1answer
28 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
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21 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
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1answer
60 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
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0answers
18 views

Norm convergence of a net of operators

Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let ...
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49 views

Region of Convergence

I know to find the Region of Convergence you find the poles of the denominator, but I'm unsure what to do for the case where there is no denominator (for a and d) and what to do if the poles are ...
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14 views

Is $X_{k+1}=\frac{1}{N}\sum_{i=1}^N \Pi_{X_{k}^{1/2}v_i}$ globally convergent?

Let $X_0=X_0^\top\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix, let $v_i\in\mathbb{R}^n$, $i=1,\dots, n$, be a set of $n$-dimensional real vectors and pick an integer $N>0$. I ...
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1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
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1answer
32 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
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1answer
25 views

Does $\mathbb{P}$-a.s. convergence preserve independence?

Let $\mathcal F$ be a $\sigma$-algebra and $X_n$ RV s.t. $X_n$ is independent of $\mathcal F$ for all $n$. Also let $X_n \to X$ $\mathbb{P}-$a.s.. Is $X$ independent of $\mathcal F$ now too?
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26 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
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5answers
40 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...
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1answer
36 views

Determine for what values $z \in \mathbb{C}$, $\sum_{n = 1}^{\infty} \frac{z^n}{n^2}$ is convergent.

I am not sure where to start on this one. I know that $z^n$ can be written as $\sum_{n=0}^{\infty} \frac{1}{1-z}$. But I do not know how to proceed.
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2answers
30 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
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71 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
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2answers
25 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
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63 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges: $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
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2answers
37 views

Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
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1answer
39 views

Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n $ converges to $0$ for a stabilised $k\in\mathbb{N}$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for ...
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27 views

Uniform continuity of this function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. We assume that $f$ satisfies the following property: For every sequence of real numbers $(x_n)_n$, there exist a subsequence $(x_{\phi ...
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26 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
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38 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
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27 views

Convergent sequences out of bounded sequences

Let us consider a bounded sequence $\{a_n\}$ . Now as it is a bounded sequence it must contain a convergent sub-sequence, $\{b_n\}$. Now let us filter out $\{b_n\}$ out of $\{a_n\}$. As such we are ...
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2answers
33 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
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0answers
28 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
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2answers
58 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
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1answer
44 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
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1answer
57 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
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3answers
21 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
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0answers
48 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
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139 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $ if \ 0 < |x-a| < \delta$ Why can't we weaken the assumption ...
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1answer
29 views

Convergence of the sequence $z_1=\frac{3}{2}$ with $z_{n+1}=\sqrt{3z_n-2}$

Prove the convergence of the sequence $(z_n)$ such that : $$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$. Calculate also the limit. I have applied induction: $$\frac{3}{2} ...
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1answer
74 views

$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=$?

$$\lim_{n \to \infty }\int_{0}^{n}\frac{n \cdot e^{\frac{x}{n}}}{x^4+n^2}dx=?$$ I am allowed to used all the classical techniques of calculus, and this was a question from measure theory when we were ...
5
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7answers
151 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
0
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1answer
33 views

Find the limit of $P_{\theta_n}\Big(\sqrt{n}(T_n-\mu(\theta_n))<z_\alpha \sigma(0)-\sqrt{n}(\mu(\theta_n)-\mu(0))\Big)$

Assumptions: Consider a sample of i.i.d random variables $X_i$ $i=1,...,n$, where each $X_i$ is defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X_i:\Omega\rightarrow ...
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1answer
23 views

How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
2
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3answers
47 views

Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, $a_n = m$.

Suppose that $a_n$ is a sequence of integers. Prove that $(a_n)$ converges if and only if there is a natural number $N$ big enough and an integer $m$ such that for any $n \geq N$, $a_n = m$. I'm ...
0
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1answer
26 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
0
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1answer
17 views

Convergence of two sequences conjecture

If $(a_n) \rightarrow 0 $ and $(b_n - a_n) \rightarrow 0$, then $(b_n) \rightarrow a$. I can't think of any counterexamples to this conjecture but I'm not sure how to prove it. Any help would be ...