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        Questions tagged [limits-and-convergence]

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

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        63 views

        Existence of limit in a recurrence equation: $\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}$ [migrated]

        Let be $\boldsymbol{\alpha_{n+2}=\frac{\alpha_{n}+(1/\alpha_{n+1})}{2}}$ a recurrence equation with known $\alpha_0$ and $\alpha_1$. How do you prove that $\lim_{n\to\infty}\alpha_n$ exists? Note that ...
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        29 views

        Convergence in probability and expectation [closed]

        Let $\mathbb EX$ exist and $X_n$ converges in probability to $X$. How to prove that $\mathbb EX_n → \mathbb EX$ then and only when $\mathbb E| X_n - X | → 0$?
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        1answer
        338 views

        Integrals of power towers

        Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
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        0answers
        41 views

        Convergence of a recursion in $L^1(0,1)$

        Let $\mu_0>0$, $a>0$, $b>0$, and $f(t)$, $g(t)>0$, $p(t)$ be some continuously differentiable functions over $\mathbb{R}$. I am looking for various tools to study the stability of the ...
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        36 views

        If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^*$ = $\mathfrak{M}(G)$

        Let $\mathfrak{M}(G)$ be the set of all means on $L_\infty (G)$ If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^* = \mathfrak{M}(G)$ My attempt: We know ...
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        50 views

        Limit Behavior of a Graph Iteration

        ,Let $G(V,E)$ be a weighted complete graph. Let further $\min_k(v_i)$ denote, depending on whether the context is arithmetic or set theoretic, either the set of the $k$ smallest edges adjacent to $...
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        1answer
        127 views

        Is the density of 1's in the Fibonacci word uniform?

        The Fibonacci word is the limit of the sequence of words starting with $0$ and satisfying rules $0 \to 01, 1 \to 0$. Equivalently, it is obtained from the recursion $S_n= S_{n-1}S_{n-2}$ under ...
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        vote
        1answer
        133 views

        Why is this series summable?

        Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that $\lim \sup_{k \rightarrow \infty} \frac{...
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        votes
        0answers
        53 views

        Does this definition of the Fourier intensity measure make sense?

        Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
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        votes
        2answers
        201 views

        How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?

        Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
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        2answers
        222 views

        Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

        For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that $$ \frac{P(X>c)}{P(X>c-1)}=1-o(1) $$ uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...
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        82 views

        Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$

        Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$ Let $V= \...
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        0answers
        121 views

        A limiting sequence of positive definite matrices

        Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...
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        votes
        1answer
        119 views

        Uniqueness of limits and compactness implies closure

        It is not difficult to prove that in a Hausdorff topological space every compact set is closed, and almost trivial that if in a topological space X every compact set is closed then X is T1. As ...
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        2answers
        370 views

        Value of $c$ such that $\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1$

        What is the value of $c$ such that $$\lim_{n\rightarrow\infty}\sum_{k=1}^{n-1}\frac{1}{(n-k)c+\log(n!)-\log(k!)}=1?$$ Numerically, it seems that the answer is $c=\log 2$. But I'd like to see a reason ...

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