Convergence of sequences and different modes of convergence.

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374 views

Proof of convergence in distribution of a discrete random variable

I'm working on a question on "convergence in distribution" and I appreciate if you could guide me on how to approach this question: Here is the question: Let $X_n$ be integer-valued random ...
3
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2answers
67 views

$\lim \sqrt[n]{a^n + b^n}$

I've seen some answers here for why this limit is the maximum between $a$ and $b$, but all of then included the hypotesis that $a$ and $b$ are both non negative. It was asked to show that this limix ...
2
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2answers
58 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
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2answers
52 views

convergence of sin functions absolutely for all x in $\mathbb R$ [closed]

How do I show that $\sum_{n = 1}^{\infty}2^n \sin(x/3^n)$ converges absolutely for all $x \in \mathbb{R}\;$?
3
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1answer
145 views

Show that $(X_{n},Y) \to^{\mathcal{D}} (X,Y)$ AND if $X=h(Y)$ where $h$ is a Borel function that $X_{n}\to^{P} X$

Let $X_{n}$, $X$, and $Y$ be real-valued r.v.'s all defined on the same space $(\Omega, \mathcal{A},\mathbb P)$. Assume that $\lim_{n \to \infty}\mathbb E\{f(X_{n})g(Y)\}=\mathbb E\{f(X)g(Y)\}$ ...
3
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2answers
64 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
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1answer
26 views

Counterexample: Convergence in finite dimensional distributions does not imply weak convergence

I'm working at the following exercise: Give an example of a sequence of stochastic processes $(\mathbb{X}^n)_{n\geq 1}$ such that the finite dimensional distributions converge to the finite ...
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13 views

Laurent Series for $\frac{1}{e^z-1}$-Radius

Can someone provide the radius of convergence of the Laurent Series of $\frac{1}{e^z-1}$ on $\mathbb{C} \backslash 0$? In class we reduced it to an inequality, but there seemed to be some debate ...
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1answer
76 views

Prove that $\int_E |f_n-f|\to0 \iff \lim\limits_{n\to\infty}\int_E|f_n|=\int_E|f|.$

I'm reading Real Analysis by Royden 4th Edition. The entire problem statement is: Let $\{f_n\}_{n=1}^\infty$ be a sequence of integrable functions on $E$ for which $f_n\to f$ pointwise a.e. on $E$ ...
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1answer
58 views

Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
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2answers
31 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
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0answers
45 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
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2answers
22 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
0
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1answer
27 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
2
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1answer
41 views

completely bounded maps - convergence

Let $x$ be a completely bounded map between operator spaces $W \subset \mathbf{B}(\mathcal{H})$ and $V \subset \mathbf{B}(\mathcal{K})$, where $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces, and ...
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3answers
52 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
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2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
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1answer
42 views

Convergence of function

Suppose that $V(t)$ is nonnegative continuous function ($\forall t:V(t)\ge0$). $\dot{V}(t) = -|h(t)|^2 + f(t)g(t)$ $f(t)$ is a bounded and uniformly continuous function. $g(t)$ is a bounded and ...
4
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2answers
457 views

Determine the convergence of $\sum_{n=1}^{\infty}\left(1-\cos\frac{1}{n}\right)$

I'm having trouble determining the convergence of the series: $$ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right]. $$ I have tried the root test: ...
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1answer
20 views

Bounded Almost Sure convergence implies convergence in pth mean

A book I'm reading gave the following result. If $X_n \to X $ a.s. and $|X_n|^p \le Z$ for some random variable $Z$ with finite expectation, then we have convergence in $p$th mean. I was wondering, if ...
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1answer
25 views

Show convergence in mean

Suppose $X(t)$ is a positive i.i.d. random process in discrete time with finite mean $a$. Let $Y(t)=\min \{a, X(t)\}$. Question is: (1)Can we claim that ...
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1answer
32 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
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36 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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2answers
64 views

Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $?

I have being trying to solve this convergence but with no success. Using the ratio test I have reached here: $$ a_n = \left( \frac{n}{n+1} \right)^{n^2} $$ And $$ \frac{1}{a_n} = \left( ...
2
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1answer
46 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
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2answers
52 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
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1answer
35 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
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2answers
59 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
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1answer
49 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
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53 views

Converges Or Diverges: $\sum _{n=1}^{\infty }\:e^{-\sqrt{n}}$

Converges Or Diverges: Attempt: int_1^∞ e^(-sqrt(n)) dn t = -sqrt(n); dt = -dn/(2*sqrt(n)); int -2*sqrt(n)/[-2*sqrt(n)] * e^(-sqrt(n)) dn int (2t)*(e^t) dt u = 2t; du = 2 dt; dv = e^t dt; ...
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What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
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1answer
33 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
4
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2answers
87 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
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2answers
52 views

Finding the convergence radius of $\dfrac{(n!)^k\cdot x^n}{(kn)!}$

If K is a integer positive, find the convergence radius of the series $$\sum\limits_{n=0}^{\infty} \dfrac{(n!)^k\cdot x^n}{(kn)!}$$ Any initial idea?
2
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1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
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1answer
45 views

Show that $f_n(\cdot):[0,1]\rightarrow \Bbb R$, $f_n(x)=x^n(1-x^n)$ is simple convergent, but not uniform convergent.

Not sure if I solved this correctly. Show that $$f_n(\cdot):[0,1]\rightarrow \Bbb R$$ $$f_n(x)=x^n(1-x^n)$$ is simple convergent, but not uniform convergent. How I solved it: $(1)$ ...
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1answer
82 views

Gradient descent (with line search) for convex functions viewed as alternation

I have fundamental confusion about gradient descent (with line search) and the reason it works. I try to explain my view here, and please tell me where it goes wrong. Let $f: \mathbb{R}^n \to ...
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1answer
82 views

Compute a sequence $A_n$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$

How can we compute a sequence $A_n$ of positive real numbers, such that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$? One way I can think of, is by defining $A_n\ln(A_n)=2^n$, but ...
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2answers
106 views

$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
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1answer
40 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
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3answers
63 views

Infinite series convergence test

Test the convergence of the following series: $${\sqrt{n+1}-1\over (n+2)^3 -1} +... \infty$$ (This is a problem I got on my test today, I constructed a similar series without the -1 part and showed ...
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1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
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2answers
49 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
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1answer
32 views

$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$ I know that ...
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0answers
37 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
4
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4answers
212 views

Showing limit of a sequence $0, \frac12, \frac14, \frac38, \frac5{16}, \frac{11}{32}, \frac{21}{64},…$

How do you show the convergence of the following 2 sequences? $0, \dfrac12, \dfrac14, \dfrac38, \dfrac5{16}, \dfrac{11}{32}, \dfrac{21}{64},...$ and $1, \dfrac12, \dfrac34, \dfrac58, ...
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5answers
260 views

Prove that this sequence converges to $\frac{3}{2}$

Prove directly from the definition that the sequence $\left( \dfrac{3n^2+4}{2n^2+5} \right)$ converges to $\dfrac{3}{2}$. I know that the definition of a limit of a sequence is $|a_n - L| < ...
6
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1answer
328 views

$C(X)$ with the pointwise convergence topology is not metrizable

I need to show that if $X$ is an uncountable Tychonoff space, then $C(X)$ is not metrizable. All I've been able to show so far is that that $F(X)$, the space of all functions with pointwise topology, ...
0
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1answer
106 views

If a sequence has two convergent subsequences with different limits, then it does not converge

Let $\{x_n\}$ be a sequence. Suppose that there are two convergent subsequences $\{x_{n_i}\}$ and $\{x_{m_i}\}$. Suppose that $\lim_{i\to\infty} x_{n_i} =a$ and $\lim_{i\to\infty} x_{m_i} =b$ where ...
2
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2answers
26 views

Convergence of $\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$

This question is a place to store proofs of the convergence of Euler's product formula for the gamma function: $$\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$$ which is convergent for ...