Tagged Questions

Convergence of sequences and different modes of convergence.

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Finding the value for $x$ for which the series converges

I am unable to follow one if the steps described in this question. My question is, can someone describe the part I do not understand? The full question and solution is below : Find the value for $x$ ...
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Convergence of improper integral with parameter

In my assignment I have to study the convergence of this integral: $$\int_{0}^{1} \frac{ln(1 + \sqrt{x})}{x (x^{\alpha}-1)} dx$$ with the parameter $\alpha >0$. In a neighbourhood of $x=0$ I ...
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If $\sum x_n$ converges absolutely . I'll have to show that $\sum \frac{x_n }{1+ x_n}$ converges . [duplicate]

I tried applying Dirichlet's test and Abel's test. Here I found some similar questions. But in most of the questions, they have assumed $x_n$ to be greater than 'zero' for all 'n', which allows them ...
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Showing that a function's Taylor series converges

Define \begin{align*} f(t) = \sum_{j=0}^\infty (-1)^j\binom{\frac{1}{2}}{j} t^j &= \sum_{j=0}^\infty (-1)^j \binom{2j}{j} \frac{(-1)^{j+1}} {2^{2j} (2j-1)} t^j \\ &= ...
$a_n = {{2^n}\over n}$ diverges? [closed]
How do I formally show that the sequence $a_n = {{2^n}\over n}$ diverges using a $\delta$-$\epsilon$ argument?
Continuous mapping theorem to show $g(x_n)$ to $g(x)$ not converges.
Let $X_n$ be a random variable sequence, such that $P(X_n=1)=1/n$ and $P(X_n=1/n)=1-(1/n)$. Let g be a function, such that $g(x)= 0$ if $x\le0$, and 1 if $x>0$ Show that $g(X_n)$ not converges ...