Convergence of sequences and different modes of convergence.

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Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
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1answer
27 views

An example of a complex power series. [on hold]

I am looking for a complex power series which is convergent for some $z\in\Bbb{C}$ but not absolutely convergent. In other words, $a_0+a_1z+a_2z+\dots$ is convergent but ...
2
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4answers
43 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
2
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1answer
7 views

Convergence , conditional distribution

here my short question. I saw the following conditional distribution, which converges, in a book: $\lim_{s \to \infty}P\left(\frac{X-f(s)}{g(s)}\leq x\mid X>s\right)=G(x)\ \forall x$ in the set ...
1
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1answer
15 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
2
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1answer
26 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
2
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3answers
102 views

Show that $\int^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$ converge.

Show that $$\int\limits^\infty_0\left(\frac{\ln(1+x)} x\right)^2dx$$ converge. I have utterly no clue on this integral. Please give me some hints. Thanks you.
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1answer
19 views

Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
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0answers
40 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
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3answers
48 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
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0answers
78 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
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2answers
37 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
2
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4answers
67 views

Convergence of $a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$

Show that the sequence $$a_n=1+1/5+1/9+\ldots+\frac{1}{4n-3}$$ does not converge but the sequence $b_n=\frac{a_n}{n}$ converges. I can show the first part. For the second part, will it be sufficient ...
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1answer
55 views

Power series and their inverses (radius of convergence of each)

Suppose I have a power series approximation $y$ to an invertible function $f(x)$, and I know that $y$ convergences around $x$ on an interval $(-R,R)$, $R$ being the radius of convergence. How are the ...
6
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1answer
46 views

About the $\lim_{n \to+\infty} \frac{1}{n}\int_0^1 \log(1+e^{nf(x)})\,dx$ (Rudin's exercise)

Problem (Rudin, R&CA chapter 2, no. 25) (i) Find the smallest positive constant $c$ such that $$ \log(1+e^t) \le c+t , \qquad t \in (0,+\infty). $$ (ii) Does $$ \lim_{n ...
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0answers
36 views

How to integrate scalar field over quarter torus? Infinite series does not converge.

This seems to be physics question, but the problem just concerns math. Preface If one wants to calculate the permeance $P$ of a rectangular bar: it is an easy task: $$P = \frac{\mu a b}{L} ...
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0answers
11 views

Convergence of the distribution of a GBM at a random time when time converges in probability

I have got the following question. Let $(S_t)_{t\in[0,T] }$ be a geometric Browninan motion. Consider a sequence of bounded random variables $(\tau_n)_{n\in\mathbb N}$ such that $\tau_n\downarrow ...
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1answer
31 views

Simpler way of proving series convergence?

Determine whether the following series converges $$\sum_{n=1}^{\infty} \left (\frac{n^4}{n^4 + 2}\right)^{n^5-3}.$$ I've found convergence using the root criterion in the following way. $\sqrt[n]{ ...
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$ max \{x : 0 \leq x < 1\} = ? $

As per the title, what is the maximum value: $$ \max \, \{x : x \in \mathbb R, 0 \leq x < 1\} = ? $$ This question came to me when considering the supremum metric applied to the set of functions ...
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3answers
62 views

A sequence and convergence

Let $\{x_n\}$ and $\{y_n\}$ be sequences of real numbers which converge to $\ell$ and $m$ respectively. Show that $$\lim_{n\to \infty} \frac{1}{n}\sum_{k=1}^{n}x_ky_{n+1-k}=\ell m$$ This is a ...
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1answer
41 views

Finding radius of convergence using root test

Find the radius of convergence of the following power series $$\sum_{n=1}^{\infty} \frac{2^n + 1}{n} x^n.$$ Using the ratio test, I have found that the radius of convergence is $R = \frac{1}{2}$. I ...
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2answers
124 views
+100

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
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0answers
26 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
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0answers
32 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
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2answers
60 views

Convergence of Sequence $a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$

Apply Cauchy's principle of convergence to prove that the sequence $\langle a_n\rangle$ defined by $$a_n=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3n-2}$$ is not convergent My attempt : consider, ...
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1answer
32 views

Iterated Functions - designing iterator to converge to constant value

I came across an interesting iterated function: $$ x_n = \frac{x_{n-1}}{x_{n-1} + b} $$ This is an extremely simple example and it converges to the constant $1-b$. Can someone provide some insight to ...
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1answer
47 views

Weak convergence of distribution family

I know the convergence in distribution and the weak convergence. but I have two questions: First one: does weak convergence implies pointwise convergence or it is the same? And second one: I have ...
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32 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
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3answers
84 views

Power Series Convergence comparison

Given $\sum_{n=0}^\infty c_n4^n$ is convergent, can this be used to find the convergence of $\sum_{n=0}^\infty c_n(-2)^n$?
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2answers
32 views

An $L^1$ convergence problem

Is the following true? If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$? ...
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2answers
85 views

How to tell if a log series converges?

I have the following series. $$(-1)^n \times \ln\Bigg(\frac{8n+5}{7n+3}\Bigg)$$ I tried the root, ratio and integral tests, but am doing something wrong because I am unable to tell if this series ...
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1answer
44 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb ...
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1answer
30 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
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1answer
33 views

A question about the convergence of series.

If$\hspace{0.3cm}$ $\sum a_n$ and $\hspace{0.3cm}$$\sum b_n$$\hspace{0.3cm}$ are convergent then which of the following is true? $1.$$\hspace{0.3cm}$$a_{n+1}<a_n$$\hspace{0.3cm}$ $\forall n$ ...
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1answer
40 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
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0answers
86 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
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1answer
117 views

Conditions for convergence of moments

Let ${X_n}$ be a sequence of r.v. such that $X_n\xrightarrow [d]{}X$, with $E(X)$ finite, and with $E(|X_n|^{1+\delta})\leq K<\infty$ for all $n$. We know that: a) For $\delta>0$, we have ...
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1answer
26 views

Norms and convergence in $\mathcal{C}^{\infty}(O)$

Let $O \in \mathbb{R}^d$ be open, $K \subseteq O$ compact and $n \in \mathbb{N}$. For $f \in \mathcal{C}^{\infty}(O)=\mathcal{E}(O)$ we define $$\|f\|_{n,K}^{(1)}:=\sup_{|\alpha| \leq n} \|D^{\alpha} ...
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28 views

Iterative Mean, Covariance Algorithm Convergence

The problem is to show that the following iterations converge to the vector $\mu$ and the matrix $\Sigma$. We have data in the form of nx1 vectors $\mathbf{Q}_k$, $1 \leq k \leq N$ where ...
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0answers
35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
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1answer
35 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
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2answers
38 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
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0answers
18 views

Convergence of the empirical characteristic function

I have to prove that the empirical characteristic function of a sample of i.i.d. random values converges in distribution against the normal distribution. I think I have to use the Multidimensional ...
0
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1answer
21 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
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semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
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1answer
55 views

for which value of $a$ that$\sum_{n=0}^{\infty} \frac{1}{u_{n}^{a}}$ converges?

We are given an arbitrary real positive $u_0$. The sequence $\{u_n\}_{n\ge 0}$ is defined by $u_{n+1}=u_ne^{-u_n}$ for $n\ge 0$. Find the values of $a\in\mathbb{R}$ for which the sequence ...
5
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4answers
107 views

Is $\sum\limits_{n=1}^\infty \sin{\frac{(-1)^{n+1}}{n}}$ convergent?

$$ \mbox{Is}\quad \sum_{n=1}^\infty \sin\left(\left[-1\right]^{n + 1} \over n\right) \quad\mbox{ convergent ?.} $$ $$ \mbox{I know that }\quad\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}\quad \mbox{is ...
4
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3answers
223 views

How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]

Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.
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0answers
50 views

Find the limits of the convergent subsequences

Let my sequence be $a_n=n\pi-\lfloor n\pi\rfloor$ This sequence is bounded in $[0,1)$ so if must have a convergent subsequence. In fact, it seems to me like it has infinitely many convergent ...
1
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1answer
18 views

weak convergence of probability measures and unbounded functions with bounded expectation

Assume that $\mu^n$ are probability measures on $R$ that convergence weakly(-*) to $\mu$, i.e for all $f \in C_b (R)$ (bounded and continuous), we have that $\int f(x) \mu^n(dx) \rightarrow \int f(x) ...