Convergence of sequences and different modes of convergence.

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59 views

Uniqueness of the Convex Combination of Positive-Definite Matrices

I am trying to connect the matrices $X$ and $Y$ with a curve defined by the convex combination of $X X^T$ and $Y Y^T$. If I define $Z Z^T = c(X X^T) + (1-c) (Y Y^T), \ c \in [0,1]$, it is true that ...
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1answer
37 views

Determine whether a function series is uniformly convergent

Determine whether $\sum_{j=0}^{\infty} \frac{\sin(jx)}{(2+x^2)^j}$ is uniformly convergent for $x\in\mathbb{R}$ So I started by saying as $|\sin(jx)|\le1$ so $\sum_{j=0}^{\infty} ...
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1answer
26 views

Asymptotic Normality of one-sample U-statistic in van der Vaart

I have a question related to the proof of Theorem 12.3 (asymptotic normality of the one-sample U-statistic) in Van der Vaart "Asymptotic Normality" here. My question is related to the point in ...
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0answers
53 views

convergence of series over $f(n+{x\over a})$ [duplicate]

let $f: \mathbb R\rightarrow \mathbb R$ be integrable and $a>0$. I have to show that for almost all $x \in \mathbb R$ the series $\sum_{n \in \mathbb Z} f(n+{x\over a})$ converges, but I have no ...
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1answer
14 views

Subsequence proof of the continuous mapping principle for convergence in probability

Theorem. Let $f$ be a continuous function. If $X_n \to_p X$, then $f(X_n) \to_p f(X)$. ($\to_p$ denotes convergence in probability.) The proof I am looking at goes Let $\{f(X_{n_k})\}$ be an ...
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8answers
129 views

Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

The following series $$\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$$ converges. It fails the divergence test, but once I apply the ratio test, the limit is always equal to $\infty$. Unless you cannot ...
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29 views

Is this sequence uniformly convergent??

I'm just going through a few practice papers and I came across a problem I am having some trouble with so would appreciate some help. The function is $\frac{\log(x)}{x^n}$ for $x\geq 1$. Now I know ...
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1answer
24 views

Prove a functional series is pointwise and/or uniformly convergent

Determine whether the following sequence is pointwise and/or uniformly convergent $(f_n)_{n\in\mathbb{N}}$ where $x\in\mathbb{R}$ and $$f_n(x)= \left\{ \begin{array}{ll} n & ...
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1answer
30 views

Slutsky's theorem with convergence in probability

Consider two sequences of real-valued random variables $\{X_n\}_n$ $\{Y_n\}_n$ and a sequence of real numbers $\{B_n\}_n$. Suppose we have that $$ \frac{X_n}{B_n}-\frac{Y_n}{B}\rightarrow_p0 $$ and ...
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2answers
48 views

Given series $a_n$ converges absolutely, then how to prove series $\log(1+a_n^4)$ converges

Given series $a_n$ converges absolutely, then how to prove series $\log(1+a_n^4)$ converges. Which test should I use? Thanks for helping.
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1answer
33 views

Division between power series that converge at least for |x| < r, only valid for |x| sufficiently small?

I have this book Calculus, Ninth Edition by Varberg, Purcell, and Rigdon; there's a particular point of a theorem (and another line after that) about Infinite Series that I really don't understand. I ...
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1answer
34 views

Show a sequence of functions is pointwise but not uniformly convergnt

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a bounded function with the properties $f(0)=0$ $\lim_{x\to\infty} f(x)=1$. For each $n\in\mathbb{N}$ define $f_n(x)=f(x+e^n)$ with $(x\in\mathbb{R})$ ...
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1answer
22 views

Is the sequence pointwise convergent

$\left(f_{n}\right)_{n\in\mathbb{N}}$ where $$ f_n\left(x\right) = \begin{cases} n, & \text{if } x\geq n \\ 1, & \text{ if } x < n \end{cases} $$ I need to determine whether the above is ...
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1answer
30 views

Relation between $L^2(\mu)$ convergence and $L^2(\nu)$ convergence.

Here's the setting: Let $\mu$ and $\nu$ be two probability measures on $(\mathbb{R},\mathcal{B})$. Assume $\mu$ and $\nu$ are equivalent (that is, $\nu\ll\mu$ and $\mu\ll\nu$). Let $(f_n)$ be a ...
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2answers
48 views

How come it be $\frac{3}{2}A$ and not only $A$?

OK I admit I was too lazy to type this question so I took a screenshot , I got it from the site @brilliant.org where it asked in terms of $A$ what would be the 2nd summation equation ? The explained ...
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1answer
43 views

Determine whether the functional series $\sum\frac1{(x+j)^2}$ and $(\sin\frac{x}{j})^j$ are pointwise and/or uniform convergent

Determine whether the following functional series is pointwise and/or uniformly convergent 1) $\sum_{j=1}^{\infty} \frac{1}{(x+j)^2}$ with $(x>0)$ 2)$\sum_{j=1}^{\infty} ...
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1answer
56 views

Is $f_n$ uniformly convergent?

Prove that $f_n(x)=(\sqrt{x^2+\frac{1}{n}})_{n\in\mathbb{N}}$ $(x\in\mathbb{R})$ is pointwise convergent, and then check to see if its uniformly convergent So I can prove it is pointwise: ...
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1answer
36 views

Infinite sum of a product of convergent sequences

Given a sequence $x_i >0 $ whose sum is absolutely convergent, i.e. $\sum_i x_i < \infty$ and another convergent sequence $0 < y_i \rightarrow y^*$, is it true that $$\sum_i x_i y_i < ...
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How many sequences of rational numbers converging to 1 are there?

I have a problem with this exercise: How many sequences of rational numbers converging to 1 are there? I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here ...
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1answer
36 views

Prove a function is pointwise convergent

Prove the following function is pointwise convergent, and following this prove further whether or not it is uniformly convergent: $(e^{-nx^2})_{n\in \mathbb{N}}$ $ (x\in\mathbb{R})$ I ...
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1answer
40 views

Mean of a Reciprocal of a Normal Distribution

I have a random variable, $\eta$, with a normal distribution with zero mean and unity variance. I process this through the function $$f(x)=\frac{1}{1-x}$$ I thought I should be able to replace this ...
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1answer
28 views

Test for convergence of integral

Considering the behaviour of the integrant at both integration limits, study the convergence of the integral: $$\int_{0}^\infty x \sin\left(\frac{1}{x^\frac{3}{2}}\right). $$ I was trying to ...
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36 views

Convergence of Sum Involving Double Factorial

I have the sum $$\sum_{I=1}^\infty a^i (2i-1)!!$$ where $!!$ is the double factorial (the product of all the integers from 1 up to some non-negative integer n that have the same parity as n is called ...
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0answers
40 views

Is this sequence of function dominated?

Let $$f_n(t) = \left\{ \begin{array} {lr} t^k e^{-t} \left( 1 - \frac {e^{-t}} n \right)^{n-1}, & t > -\log n \\ 0, & t \le -\log n \end{array} \right.$$ for $k=1,2$. Is there an integrable ...
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1answer
49 views

Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space?

Suppose $<M,d>$ is a metric space. Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space? Or maybe lim$_{a \rightarrow b } \space d(a,b) \neq 0 $ implies ...
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2answers
27 views

prove that $x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$ is cubic order of convergence near $x_0=\sqrt{5}$

To solve the equation $$x^2-5=0$$ There exitsts a iteration method $$x_{n+1}=\frac{x_n(x_n^2+15)}{3x_n^2+5}$$ I know that it is cubic convergence but I don't know how to prove it. I have tried the ...
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27 views

Radius of convergence of $\frac{x}{sinh(x)}$

The power series representation of real hyperbolic sine function, as $sinh(x)= \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$. And its radius of convergence is, of course, $\frac{1}{\lim_{n\to \infty} ...
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53 views

Convergence, find the mistake I made!

I admit it: I am a student, and this is my homework. Still, I am at the end of my wits. I have tried solving it like this: There is a mistake somewhere. I cannot find it. All I am asking is to give ...
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102 views

Nested Radicals Involving Primes

How do you evaluate $\sqrt { 2+\sqrt { 3+\sqrt { 5+\sqrt { 7+\sqrt { 11+ \dots } } } } } $ ? This question appears to be rather difficult as there is no way to perfectly know what $p_{ n }$ is , ...
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1answer
26 views

Confusion in not extending results in convergence in probability to convergence in distribution

We know that if $X_n\xrightarrow{p}X$ and $Y_n\xrightarrow{p}Y$, then $X_n+Y_n\xrightarrow{p}X+Y$, and since $X_n\xrightarrow{p}X\Rightarrow X_n\xrightarrow{d}X$ (and simliarly for $Y_n$), why does ...
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70 views

The series of $\sin A$, when $A$ is a $n \times n$ matrix

I have to solve this problem for university. $\sin A$ is defined for every $n \times n$ matrix $A$ as: $$\sum_{n=0}^\infty \frac{(-1)^n A^{2n+1}}{(2n+1)!} \tag{$\star$} $$ Prove that a) $\| \sin ...
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1answer
30 views

Same integral diverges for different limits?

I was investigating convergence of such an integral: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ I used comparison test: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ < $\int_{1}^\infty $$\frac{dx}{x^2} $ ...
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1answer
54 views

How to determine the limit of a sequence in a metric space

If I'm trying to prove that a Metric space $(M,d)$ is not complete I have to find a Cauchy a sequence that doesn't converge in $M$. Using the following Metric Space as an example: $M = \{ x \in ...
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2answers
78 views

Is $\sum_{n=1}^\infty\frac{(-1)^n}{n}$ convergent?

According to Leibniz's standard, $$\sum\limits_{n=1}^\infty\dfrac{(-1)^n}{n}$$ converges, but what about $$\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n}$$ Does it converge?
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46 views

Does the sequence have a convergent subsequence?

I'm trying to figure out if the sequence $e^{(-n)^n}$ where n is a natural number has a convergent subsequence? It's in a past exam paper. I know that obviously I can't apply the Bolzano-Weirstrass ...
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2answers
40 views

Proof involving weak and weak-* convergence

"Prove that if $(f_n)_{n=1}^\infty\subset X'$ converges strongly in $X'$ and a sequence, $(x_n)_{n=1}^\infty\subset X$ converges weakly in $X$, then, $f_n(x_n)\to f(x),\,n\to\infty$." My attempt: ...
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0answers
25 views

convergence of a decreasing sequence [duplicate]

I have problems in finding the convergence of a decreasing sequence. Suppose we have $\{x_t\}^T_{t=1}$, and it is decreasing sequence with $\lim_{t\rightarrow\infty}x_t = x^*$. Now I would like to ...
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1answer
50 views

Show $\lim_{a \rightarrow \infty} \int_0^1 {f(x)x\sin(ax^2)}=0$.

Suppose $f$ is integrable on $(0,1)$, then show $$\lim_{a \rightarrow \infty} \int_0^1 {f(x)x\sin(ax^2)}=0.$$ I tried to write $$(0,1) = \bigcup _{k=0}^{{a-1}} ...
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1answer
93 views

Determine convergence when AST does not work

Consider this series -- $$\frac{1}{2^3}-\frac{1}{3^2}+\frac{1}{4^3}-\frac{1}{5^2}+\frac{1}{6^3}-\frac{1}{7^2}+\frac{1}{8^3}-\frac{1}{9^2}+\dots$$ Apparently, alternating series test (AST) cannot be ...
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1answer
40 views

When the Plethystic sum converges?

We have \begin{align} & \exp\left(-\sum_{p=1}^{\infty} \frac{x^p}{p}\right) = 1-x, \\ & \exp\left(-\sum_{p=1}^{\infty} \frac{\sum_{k} x_k^p}{p}\right) = \prod_{k}(1 - x_k). \end{align} When ...
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5answers
91 views

How to resolve this absolute value inequality $|1+x^2|>|x|$?

I am stuck trying to find all x that satisfies $$|1+x^2|>|x|$$ ($x=0$ is obvious.) To provide more context, I had applied the root test on the series $$\sum_{n=1}^{\infty} \frac{nx^n}{n^2 + ...
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0answers
13 views

An counterexample about different convergence modes

Can someone explain to me why this holds: Let $X=[0,\infty)$, $f_n(x)=\begin{cases} 1-n\cdot(x-k) &\text{ if } k\leq x\leq n^{-1}+k\\0 &\text{ if } n^{-1}+k\leq x<k+1.\end{cases},~~~ for ...
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Does the limit of the sum equal the sum of the limits in this case?

I'm interested in evaluating the following limit: \begin{align*} \lim_{N\to\infty}\sum_{n= 1}^N\frac{1}{N\sin\left(\frac{\pi n}{N}\right)} \end{align*} Because \begin{align*} ...
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0answers
47 views

Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
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3answers
69 views

For what values of x is $|x(1-x)| < 1$?

The series is: $$\sum_{n=1}^{\infty} \frac{x^n(1-x)^n}{n}$$ I used the ratio test to try and find the range of values of $x$ for which a series converges, and am stuck with $$|x(1-x)| < ...
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2answers
37 views

Laurent Series computation

Need to compute the Laurent Series: $$f(z) = \frac{1}{1+z^2} \quad on \quad 0<|z-i|<2$$ $$f(z) = \frac{1}{2i(z-i)} + \frac{i}{2(z+i)}$$ $$\frac{1}{2i(z-i)} = \frac{1}{2zi}\sum_{k=0}^\infty ...
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2answers
41 views

Why does this function converge point-wise to $0$?

Let $$f_n(x) = \begin{cases} \sin nx & 0 \leq x \leq \frac\pi n\\ 0 & x \geq \frac\pi n \end{cases}$$ Then my book says that $f_n \to f \equiv 0$ on the interval $[0, +\infty)$. I don't ...
2
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1answer
35 views

Converge properties of Taylor Series expansion of complex function

I need to find the convergence properties of the Taylor Expansion of $$f(z)=\frac{z}{z-1}$$ I found the Taylor Series: $$\sum_{j=1}^\infty \frac{(-1)^{j+1}(z-i)^{j-1}}{(i-1)^j}$$ Then I used the ...
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0answers
20 views

Analytical algorithm to obtain solution to convex optimization problem.

Assume a vector $\vec{P}$ with N elements $\in \mathbb{R}^+$ and constants $T_P$ and $\epsilon$. The vector $\vec{P}$ is arranged in a column $(N\times 1)$. Consider the problem: $$ \begin{aligned} ...
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0answers
19 views

Uniform convergence of a functional series.

$$\sum_{j=1}^{\infty}f_j \text{ where } f_j(x) = \begin{cases} \frac{1}{j} & x\in (j-1,j] \\ 0 & x\notin (j-1,j] \end{cases} x\in\mathbb{R}$$ I have found the functional ...