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Questions tagged [convergence]

Convergence of sequences and different modes of convergence.

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Convergence of a constant sequence in a set of finite elements

In Munkres Topology section 16 in the subsection on Hausdorff Spaces there is a motivating example involving the three-point set $\{a,b,c\}$ which states that the sequence defined by setting $x_n=b$ ...
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0answers
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Limit of derivative of moment generating function

Suppose $X>0$ is an integer-valued with $P(X=k)=q_ke^{-t_0k}$, where $t_0>0$ and $\{q_k\}$ satisfies $\frac{1}{k}\log q_k \rightarrow 0$. Let $\phi(t)$ be the MGF of $X$. Let $t_{\text{max}}=\...
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1answer
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Almost sure convergence and bounded supremum

Suppose $\lim_{n→\infty} X_n=X$ a.s. and $|X|<\infty$ a.s. Let $Y=\sup_n|X_n|$. Show that $Y<\infty$ a.s. I stumbled across this problem while reading through Jacod and Protter's Probability ...
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1answer
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Is convergence of a net of functions almost everywhere a topological notion?

There exists no topology on the set of Lebesgue-measurable functions such that a sequence is convergent in that topology if and only if it is convergent almost everywhere (i.e. everywhere except for a ...
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Probability Convergence Proof [on hold]

Assume that for any $i\geq 1$, $P(A_i )=1$. Show that $P \Bigl( \bigcap _{i=1}^\infty Ai \Bigl)=1 $
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Prove that a function which converges pointwise to a limit also converges uniformly.

In general, pointwise convergence does not imply uniform convergence, but is the following proposition true (I think it is?) and how does one prove it? If $ f_n: \mathbb{N} \to \mathbb{R} $ is a ...
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How many terms it takes for the Leibniz series to converge to three decimal places of accuracy?

I need to find out how many terms it takes for the this series to converge to three decimal places of accuracy of Pi. e.i how many it terms it takes to obtain the value 3.141 from, the series: Leibniz ...
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1answer
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If $D$ is a ultrafilter on $I$ and $(a_i) \mapsto_{D} a$ and $f_i \mapsto_{D} f$ then $ Sup_{x}f_i(x, a_i) \mapsto_{D} Sup_{x} f(x, a)$

Let $X$ be a topological space and let $(x_i)_i \in I$ be a family of elements of $X$. If $D$ is an ultrafilter on $I$ and $x \in X$, we write $$x_i \mapsto_{D} x$$ if for every neighborhood $U$ of $x$...
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prove $\sum |{x_n}|$ converges, then $\sum {x_n}$ converges

I need to prove that $\sum |{x_n}|$ converges, then $\sum {x_n}$ converges too. However, isn't this false and the converse is true instead?
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Proving that a series is uniform convergent, given a certain series of functions

Given are two sequences $(f_k)_{k\geqslant 0}$ and $(g_k)_{k\geqslant 0}$ of functions $[0,1]\rightarrow\mathbb{R}$. Also it is given that $\left|f_k(x)\right|\leqslant g_k(x)$ for all $k\in\mathbb{...
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Iterative Vector Matrix Product

Consider the following iterative matrix product: $x_{i+1} = A_{i} \cdot x_{i}$. The matrices $A_i$ are defined as follows: All rows except of the first and last row are stochastic. The first and the ...
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2answers
25 views

Convergence of a sequence for the three-point set

In Munkres Topology section 16 in the subsection on Hausdorff Spaces there is a motivating example involving the three-point set $\{a, b, c\}$ which states that the sequence defined by setting $...
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0answers
27 views

Recurrence Relations with non-constant coefficients

Consider the following recurrence relations: The equations for x and y are similar and connected to each other at $x_1, y_1, x_n$ and $y_n$. $x_1(t) = \frac{1}{2\sqrt{x_1(t-1)^2 + y(t-1)^2}} x_1(t-1) ...
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Proving a function is continuous but unbounded

Problem: Hi, I'm working on the following math problem: Suppose a set $S$ contains a sequence that converges to a point $x_{0}$ that is not in $S$. Show that the function $f : S \rightarrow \...
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1answer
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Is $(L^q(E),\|.\|_p)$ a Banach space?

Let $E$ be a measurable subset of $\mathbb R$ with finite measure and let $1\leq p<q<\infty$. We know that $L^q(E)\subset L^p(E)$. Can we say that $(L^q(E),\|.\|_p)$ a Banach space? I feel there ...
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Why $f_n(x)=\chi_{[n,n+1]}$ not converges almost uniformly in $\mathbb{R}$

Let $(\mathbb{R},\mathbb{B}, \lambda )$ the real line with Lebesgue measure on the Borel Subsets of $\mathbb{R}$. why the sequence $f_n(x)=\chi_{[n,n+1]}$ not converges almost uniformly in $\mathbb{...
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1answer
31 views

Two definitions of continuity

I just want to check whether these proofs are correct. The problem goes as follow. Given $$g(x) = \begin{cases} \sin\frac{1}{x} & x>0 \\ 1 & x\leq 0 \\ \end{cases},$$ prove that $g(x)$ is ...
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4answers
37 views

Show that a recursive sequence $(x_n)$ is Cauchy

The given sequence is defined as $x_1 = 0$, $x_2 = 1$ and $x_{n+2} = \frac{1}{3}x_{n+1} + \frac{2}{3}x_n$ for $n \geq 1$. I seek to show that it is Cauchy. So how I planned on showing this was to ...
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How does $\exists N\; \forall\varepsilon>0 \;\forall k > N \; (d(x_n,x) < \varepsilon)$ differ from definition of convergence?

Show $\exists N\; \forall\varepsilon>0 \;\forall k > N \;(d(x_n,x) < \varepsilon) \qquad \implies \qquad\forall\varepsilon>0\;\exists N\;\forall k > N \;(d(x_n,x) < \varepsilon)$ ...
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Determine whether this is true: If $a_n \rightarrow +\infty$ then $\sum_{n = 1}^{\infty}\frac{1}{a_{n}^{n}}$ converges

Determine whether the following is true: If sequence $a_n \rightarrow +\infty$ then the infinite series $\sum_{n = 1}^{\infty} \frac{1}{a_{n}^{n}} $ converges. Note: I am looking for feedback on ...
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Convergence of i.i.d. random variables with respect to index

This is exercise 2.3.17 in Probabilty:Theory and Example p67. $X_1,X_2,X_3\cdots$ are i.i.d random variables. The reader is reqeusted to find the equivalent condition for each of the four statement. ...
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2answers
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How to check if $\sum_{n=1}^\infty \frac 1{n!}$ is converging or diverging by direct comparison test?

$$\sum_{n=1}^\infty \frac 1{n!}$$ I think we may compare it with $\frac{1}{n^n}$, and say that $\frac{1}{n!}>\frac{1}{n^n}$. But I am not sure of the convergence or divergence of $\frac{1}{n^n}$ ...
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1answer
19 views

Is convergence of elementary (simple) functions always uniform?

Say I have a bounded continuous function $g(t)$ on the interval $[a,b]$ Let $$g_{n}(t)=\sum_{j=1}^{n}g(t_{j})X_{[t_{j},t_{j+1}]}(t)$$ Where X is the characteristic function, and $t^{n}_{j}$ a ...
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2answers
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When can we say that something is equal to, rather than something approaches a limit?

As an example, if we have a binary term $x$, like this $x = 0.d_1 d_2 d_3 \dots$ Where $d_1 = 1$ if $x < \frac{\pi}{ 10}$ else $d_1 = 0$ $d_2 = 1$ if $x < \frac{\pi}{ 10}$ else $d_2 ...
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2answers
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Proving $\sum_{n=1}^\infty \frac{\cos \pi n}{2^n}$ converges by the comparison test [on hold]

$$\sum_{n=1}^\infty \frac{\cos \pi n}{2^n}$$ How can I show whether or not the following series converges using the comparison test?
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0answers
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Is a power series with integral coefficients with a limit at x=1 a polynomial?

If we have a power series with integral coefficients $f(x)=\sum_{i=0}^{\infty}a_ix^i$ which converges for all $|x|<1$, and the limit as $x\rightarrow 1$ exists, where we approach along the real ...
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convergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$.

I am trying to check the convergence or divergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$. My attempt: for a finite $p$,\begin{align}\displaystyle\...
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0answers
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Find the radius of convergence R for power series $\sum_{i=0}^{\infty} n^a z^n$

For power series $\sum_{i=0}^{\infty} n^a z^n$ The series seems to be diverging as $C>1$. How can I find the radius of convergence R at this instance?
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1answer
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Equivalence of conditions for Convergence of Non-negative Random Series

Question Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent. $\sum_{n=1}^\infty X_n<\infty$ a.s. $\sum_{n=1}^\infty[P(X_n>1)+E(X_nI(X_n\leq 1)]<\infty$ $...
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1answer
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Prove that the following set $A = \{x \in \mathbb{R} \setminus\mathbb{Q}\mid x^2 \leq \frac{1}{4} \} \subset \mathbb{R}$ is complete.

A subset $S$ of $\mathbb{R}$ is complete if every Cauchy sequence consisting of elements of $S$ converges to an element of $S$. Prove that the following set $A = \{x \in \mathbb{R} \setminus\mathbb{Q}...
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3answers
53 views

Does the series $\sum\limits_{n=1}^{\infty} (2^{1/n} - 2^{1/(n+1)})$ converge?

I need to argue if this series converges or diverges. I know that $$\sum_{n=1}^{\infty} 2^{\frac{1}{n}} = \infty$$ and $$\sum_{n=1}^{\infty} -2^{\frac{1}{n+1}} = \infty$$ But my first impression ...
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1answer
22 views

Given the series (-1)^n.tan(1/n) how do I study its nature in terms of divergence and convergence?

I have a series whose general term is tan(1/n)*(-1)^n and I want know if it is divergent or convergent, how do I proceed? I have tried stablishing upper and lower limits and the ratio test and all I ...
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2answers
25 views

proving convergence by showing a sequence is Cauchy

Let $(x_n)_{ n=1}^{∞} $ be a sequence of real numbers such that $|x_{n+1} − x_n| ≤ \frac {1} {2^n}$ for all $n$. Show that $(x_n) ^∞ _{n=1}$ converges. It seems pretty clear that I can prove that ...
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4answers
63 views

Prove that the following set $A=\{2n−1∥ n∈N\} \subset \mathbb{R}$ is complete.

A subset $S \subset \mathbb{R}$ is complete if every Cauchy sequence consisting of elements of S converges to an element of S Prove that the following set $A=\{2n−1∥ n∈N\} \subset \mathbb{R}$ is ...
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2answers
24 views

Which criterion do we use for the convergence of the sequences?

I want to check as for the convergence the following sequences: $$\left( \left( 1+\frac{1}{n}\right)^{n^2}\right), \left( \left( 1+\frac{1}{n^2}\right)^{n}\right), \left( \left( 1+\frac{1}{\sqrt{n}}\...
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2answers
115 views

Convergence of series $\sum\int\limits_{1}^{+\infty}e^{-x^n}\,dx$

Determine if the series $\sum \alpha_n$ converge, where: $$\alpha_n=\int\limits_{1}^{+\infty}e^{-x^n}\,dx.$$ Attempt. Ι am pretty sure the inequalities $$e^{-x^n}\leq \frac{1}{1+x^n}$$ and $e^{-...
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5answers
63 views

Determine if $\sum_{n=0}^{\infty}(-1)^n\frac{n+1}{n^2+1}$ converges or diverges

I'm having trouble figuring this one out. $$\sum_{n=0}^{\infty} (-1)^n\frac{n+1}{n^2+1}$$ I think this is conditionally converging as it has $(-1)^n$ so we should take $\lvert(-1)^n\rvert$? I'm a ...
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4answers
66 views

Determine if $\sum_{n=0}^∞ \frac{n}{(2i)^n}$ is convergent, divergent

$\sum_{n=0}^∞ \frac{n}{(2i)^n}$ I have been trying to work out how to do this as this involves $i$, usually we would use ratio test but I wasn't sure how to. How would you work this out with ...
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1answer
20 views

A question about a.s. convergence

While reading on almost sure convergence, I came across this equivalence: $$\{\omega \in \Omega: \lim_n X_n(\omega) = X(\omega)\} \equiv \bigcap\limits_{k=1}^{\infty}\bigcup\limits_{N=1}^{\infty} \...
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0answers
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Prove that $\frac{k_1\alpha_1+\ldots+k_n\alpha_n}{k_1+\ldots+k_n} \to \alpha$ for $\alpha_n \to \alpha.$ [duplicate]

If $(k_n)$ is a sequence of $\mathbb{N}$ and sequence $(\alpha_n)$ converges to $\alpha\in \mathbb{R}$, prove that the convex combination: $$\frac{k_1\alpha_1+\ldots+k_n\alpha_n}{k_1+\ldots+k_n} \...
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Upper bound of a sequence for all n

Let $r_n$ is a non-increasing sequence such that $ 1 \leq r_n \leq 1 + \theta_1^2 + \theta_2^2$ and $r_n \rightarrow 1$ as $n \rightarrow \infty$. Further we know that $|\theta_2| < 1$ and $|\...
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2answers
44 views

Show that $\sum_{n=1}^{\infty}\frac{\log (1+1/n)}{n}$ converges

I need some help here. I can show that if the Cauchy condensation test holds, then I get two separate series, one which converges by the comparison test, and one that converges by the ratio test. But ...
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1answer
20 views

If a series converges then any series with subsequence of terms converges also

I tried to look if this was already answered on here. Couldn't find it, but maybe I am just not searching well. Let $\sum_{n=1}^\infty a_n$ be a series with nonnegative terms which converges. Let $\{...
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0answers
34 views

Geometric mean of a sequence converges

Let $\{a_n\}$ be a sequence of positive numbers such that $\lim_{n\to\infty} a_n = L$. Prove that $$\lim_{n\to\infty}(a_1\cdots a_n)^{1/n} = L$$ Proof: Let $\epsilon > 0$. There exists $N\in\...
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2answers
20 views

Given a sequence $(a_{n})$, what does it mean for it to have a convergent subsequence (say to real number L)?

Is there any algebraic way to represent a subsequence converging to L? Let the subsequence be denoted by $(b_{n})$, then does this imply for all $n\geq N, \epsilon \gt 0 $, we have that: $\lvert b_{n} ...
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1answer
45 views

Does $X_n \to X$ in probability imply $\mathbb{E}(\liminf_n X_n) = \mathbb{E}(X)$?

My question is about the relation bwtween pointwise limit and convergence in probability. It seems a basic question but I am stuck to it. Suppose $X_n \rightarrow X$ in probability Then we konw $\...
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0answers
31 views

Show that $\sum_{l=0}^{\infty}(\gamma\matrix{A})^l$ converges

Based on the following conditions where $\matrix{A}$ is a matrix: $\left| \lambda\matrix{A} \right|=\left| \lambda \right|\left| \matrix{A} \right| $ for any $\lambda \in \mathbb{R}$ $\left| \matrix{...
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1answer
34 views

Power series radius of convergence and if they are divergent or not

$\sum_{i=0}^{\infty} e^{-\sqrt n}z^n $ I tried to find the radius of convergence of a power series.. is this equation a geometric series? or would it be easier to do a ratio test and $\lim_{n\to\...
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1answer
17 views

Proving Convergence of a series using the Cauchy or d'Alembert test

How do I go about proving that $$\mathbf\sum _{n=1}^{\infty }\:\left(\frac{1}{2^n}+\frac{1}{3^n}\right)$$ converges? Using the ratio test: $$\frac{|a_n+1|}{|a_n|}<1$$ I was able to simplify the ...