Convergence of sequences and different modes of convergence.

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2
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1answer
117 views

if $ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$ then $\sum_{n=1}^\infty a_n = L$

let ${a_n}$ be a sequence of Real non negative numbers. assume the following limit exists and is finite: $$ L = \lim_{x\to1^-} \sum_{n=1}^\infty a_nx^n$$ prove that $\sum_{n=1}^\infty a_n$ ...
0
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0answers
23 views

A problem upon function series

Function series $\sum_{n=1 }^{ \infty} u_{n}(x)$ converges to $S(x)$ in bounded interval $[a,b]$, if every $u_n(x)$ is non-negative and continuous in $[a,b]$. prove that $S(x)$ attains its infimum in ...
5
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1answer
46 views

Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. Prove that $f_n$ converges to $f$ in $L^1$ norm.

Let $\{f_n\}$ and $f$ be Lebesgue measurable functions on $E$ where $|E|<\infty$. Assume that $f_n\to f$ in measure and $\sup_n\|f_n\|_{L^p(E)}<\infty$ for some $p>1$. (a) Prove that ...
4
votes
2answers
91 views

If $\prod_{n}(1+a_n)$ converges, does $\sum_{n}\frac{a_n}{1+a_n}$ converge?

I have a sequence of complex numbers $a_1,a_2,...$ such that $a_i \neq -1$. I then have the infinite product $\prod_{n=1}^{\infty}(1+a_n)$ which I know converges to a non zero complex number. I was ...
0
votes
1answer
30 views

The Comparison test or The limit comparison test

I have checked many of this site pages yet I could not find a clear answer about how to choose between The "Comparison test" OR The "limit comparison test"? Because the difference between the two is ...
1
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0answers
17 views

Generalization of the Central Limit Theorem for Local Martingales

the Central Limit Theorem for Local Martingales states the following. Theorem Let $M_n = (M_n(s))_{s \geq 0}$ be a square integrable local martingale such that for all $T > 0$ $$ \lim_{n ...
6
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1answer
97 views

calculate $\prod_1^\infty k^{\frac1{k!}}$

Is it possible to find the point of convergence of $\prod_1^\infty k^{\frac1{k!}}$ $K!=k(k-1)!$. My attempt: If $S_n=\prod_1^\infty k^{\frac1{k!}}$ then $\ln S_n=\sum_1^\infty \frac{\ln k}{k!}< ...
0
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4answers
72 views

How to find the limit of a Sequence $ f(n) = \frac{16f(n-1)-2}{32} $

I have this sequence: \begin{equation} f(n) = \frac{16f(n-1)-2}{32} \end{equation} with \begin{equation} f(0) = 1 \end{equation} I want to find it's limit while n goes to infinity and therefore check ...
0
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0answers
20 views

How are the tolerances evaluated in fmincon? specific/complete mathematical formulations needed.

I'm currently studying the stopping criteria about fmincon using different algorithms and I'm wondering how are the tolerances are actually evaluated and compared in the built-in function fmincon. ...
2
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1answer
57 views

is $ f_n(x) = (1+x^n)^{1/n}$ uniformly convergent

does the following series of function is uniformly convergent in $[0,\infty]$? $ f_n(x) = (1+x^n)^{1/n}$ I found that $f_n \to f(x) = \begin{cases} x, & 1 \le x \\ 1, & 0 \le x \le 1 ...
5
votes
2answers
111 views

Computing an improper integral with respect to a parameter

I am motivated by this problem.Let us compute an improper integral with respect to a parameter:$$F(x)=\int_{1}^{\infty}\frac{e^{-xy}-1}{y^{3}}dy,\quad x\in[0,\infty).$$ The following is my ...
1
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0answers
72 views

Can I use Dominated Convergence Theorem for this sum?

$$\left[\sum_{n=0}^\infty \frac{a_n}{n!} (sz)^n\right]e^{-s}$$ Does this product decay to zero, as $s$ goes to infinity? Assume the left-hand term, call it $\phi(sz)$, is entire. Also, you can ...
0
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1answer
27 views

How to show that $P(X_{n} \geq n$ $i.o.) = 0$, given $E(X_{i}) = 0$ and $E((X_{i})^{2})=1$ for $i=1,2,3…$

I'm struggling with the following problem (Exercise 4.5.16 in Rosenthal's probability book): Let $X_{1}, X_{2},...$ be defined jointly on some probability space, with $E(X_{i}) = 0$ and ...
0
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2answers
46 views

How to show convergence in a $\mathbb{R}^n$?

I've come across a chapter in my book which has me stumped and nowhere can I find so that I can move on. The question is "Using the definition 9.1i, prove that the following limits exist: a) $$x_k = ...
0
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2answers
53 views

Convergence of a sequence by induction: $x_{n+1}=\frac{x_n+1}{x_n+a},x_1>0,a>0,n=1,2,…$

Assume that $x_n>0$ and prove $x_{n+1}>0$ $x_{n+1}=1-\frac{a-1}{x_n+a}$ $x_n+a>a$ $-\frac{a-1}{x_n+a}>-\frac{a-1}{a}\Rightarrow x_{n+1}>0$ Is it necessary to find upper bound to ...
0
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0answers
12 views

Convergence random matrix inverse

I have the following problem: A is a sum of independent random matrices that converges in expectation to say a matrix C and B is some fixed positive (semi-)definite matrix. I'm interested in a bound ...
4
votes
1answer
92 views

How do I evaluate this limit :$\displaystyle \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$?

I would like to know if this :$$ \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$$ does exist and how do i evaluate it ?. Note : I have tried to use the standard limit : $$ \lim_{z\to \infty} ...
0
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0answers
55 views

How do we evaluate this limit — a product of a series, which is entire, and a decaying exponential function,

$$lim_{s-> \infty}\phi(sz)e^{-s}$$ where $\phi$ is the series $$\sum_0^\infty \frac{a_n}{n!}(sz)^n$$ and is entire. Also, assume that we have this upper bound: $|\phi(sz)|$ $\le$M$e^{|sz|}$. ...
0
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0answers
41 views

Why does the harmonic series diverge? [duplicate]

I had a little doubt over a small matter: if $(1/n)\to0$ even as $n\to \infty$ then how come the summation $$\sum_{n=1}^{\infty} \frac 1 n$$ is divergent (as stated in my textbook).
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2answers
42 views

Expected value definitions

Let $X : \Omega \to \mathbb{R}$ be a discrete random variable in a discrete probability space with countable sample space $\Omega$. Let $P(\omega)$ be the probability of an outcome $\omega \in ...
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2answers
36 views

Get stuck in judging convergence of a improper integral

I get stuck in judging convergence of the following formula:$$\int_0^1{\frac{\sqrt[m]{\ln^2(1-x)}}{\sqrt[n]{x}}}dx$$ $m$ and $n$ are both integers When $x\to0^+$, I can use equivalent infinitesimal ...
5
votes
2answers
48 views

On the convergence of $\sum a_n \cos nx$ , $\forall x \in \mathbb R \setminus \pi \mathbb Z$ , where $\{a_n\}$ is monotone [closed]

Let $\{a_n\}$ be a monotone sequence of real numbers ; if $\lim_{ n \to \infty} a_n=0$ , then is it true that $\sum a_n \cos nx$ is convergent $\forall x \in \mathbb R \setminus \pi \mathbb Z$ ?Is the ...
4
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1answer
39 views

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$'s are disjoint. Let $f : A \to \mathbb{R}$ take elements ...
4
votes
1answer
139 views

Integrate by parts to prove that this integral provides an analytic continuation ,

Suppose $f(z) = \sum_0^\infty a_nz^n$ converges for $|z| \le 1$. a) Prove $\phi(z) = \sum_0^\infty \frac{a_n}{n!}z^n$ is entire and $|\phi(z)|\le Me^{|z|}$. b) Prove $f(z) = \int_0^\infty ...
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0answers
30 views

Implicit Fact about Bounded Monotone Sequence converges

If you have a sequence $ a_n $ which is bounded and monontonic increasing the theorem tells us it converges to a limit $ L $ But having looked at the proof is it implicit that this limit $ L= Sup ...
4
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5answers
124 views

Does this double sum $\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{m+n+mn}{2^m(2^m+2^n)}$ converge?

$$\sum_{m=0}^\infty \sum_{n=0}^\infty \frac{m+n+mn}{2^m(2^m+2^n)}$$ I have no experience evaluating double sums, but what intuition I have about single sums suggests to me that this series should ...
-1
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2answers
41 views

Proving that $a_{n}=\frac{(-1)^n*n + 1}{n+2} $is divergent

In my assignment I have to prove that the following sequence is divergent: $$a_{n}=\frac{(-1)^n*n + 1}{n+2} $$ I have an idea but I don't know if my solution is correct. Here it is: Split into ...
1
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2answers
17 views

Extending continuous function from a dense set

If $X$ is a metric space and $Y$ a complete metric space. Let $A$ be a dense subset of $X$. If there is a uniformly continuous function $f$ from $A$ to $Y$, it can be uniquely extended to a uniformly ...
0
votes
1answer
53 views

How do I evaluate this sum :$\sum_{n=2}^{\infty}\frac{3^{n}(2^{n-1}-1)}{2^{n}(3^{n-1}-1)}$?. [closed]

i would like to know how do i evaluate this sum :$$\sum_{n=2}^{\infty}\frac{3^{n}(2^{n-1}-1)}{2^{n}(3^{n-1}-1)}$$. Note :I used many creterions but i can't determine wether if it is convergent ...
0
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1answer
16 views

How can one read the order of convergence from a loglog-graph?

I am making a task which includes running a Monte Carlo simulation and calculating the order of convergence experimentally. I have to calculate (or approximate) the order of convergence using ...
1
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2answers
48 views

Splitting an infinite series

Let $A$ be a countably infinite set, and let $f:A\to\mathbb{R}$ be a function that takes elements of $A$ to the reals. Suppose that $\sum_{w\in A} f(w)$ is well-defined (see note below). Also suppose ...
2
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3answers
47 views

Divergent or convergent? [closed]

If $$A_n = \frac{n(n+3)}{(n+1)^2}$$ then is the sum of the sequence $\sum(A_n)$ convergent or divergent?
3
votes
3answers
88 views

If $f_n \to f$ , $f , f_n \in \mathcal R[a,b] $ , then is it true that $\lim_{n \to \infty} \int_a^bf_n=\int_a^b f$ ?

Let $\{f_n\}$ be a sequence of real valued functions with domain $[a,b]$ converging pointwise to $f$ and such that each $\{f_n\}$ and $f$ is Riemann integrable in $[a,b]$ , then must it hold that ...
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0answers
15 views

Doubt regarding limitting value of partial derivative of $C^{1}$ function

The aim is to prove the following result: Let $v : \mathbb R \to \mathbb R$ be such that: $v \in C^{1}(\mathbb R)$ & $|\frac{\partial v}{\partial x}| \in L^{1}(\mathbb R)$ . Then to prove that: ...
2
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1answer
31 views

Pointwise Convergence to 0 Implies Uniform Convergence to 0

I have seen some related posts on Dini's Theorem, and am actually working a problem related to it, but I have come across some troubling logic unrelated to the theorem. I believe my question to be ...
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2answers
33 views

Equivalent Criteria for convergence of a sequence

My textbook of Metric Spaces describes the following equivalent criteria for convergence of a sequence $\{x_n\}$ : $(i)~ \bigcap \{ \overline {x_n~|~n \in S}~|~S \subseteq \mathbb N, S$ infinite $\} ...
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1answer
44 views

The limit of a series of continous functions is continuous.

Given a continuous function $f_0: [0,1] \rightarrow \mathbb{R}$, define $$f_n(x) = \int^x_0 f_{n-1}(t) dt, x \in [0,1]$$ for $n=1,2,3,...$ . For each $x \in [0,1]$, show that ...
2
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1answer
39 views

Show that $a_n>0$ for all sufficiently large $n$

Let $F_n, G$ be distribution functions on $\mathbb R$. Suppose that $F_n(a_nx+b_n)\to G(x)$ as $n\to\infty$ for each $x\in c(G)$ where $c(G):=\{x\in\mathbb R:G(x)-G(x-)=0\}$. Here $a_n,b_n$ are ...
0
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2answers
54 views

For what $(a,b) \in R^+$ does $\int^\infty_b (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}})dx$ converge?

For what pairs $(a,b) \in R^+$ does this integral converge? $$ \int\limits^{\infty}_{b} \left (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}} \right)dx $$
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2answers
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find the interval of convergence of the power series

like the title said i have to find the interval of convergence of this power series : $$\sum_{n=1}^\infty{ ((-1)^n *(x-1)^{2n-1})\over 3^n}$$ I applied the ratio test and i got something ...
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0answers
37 views

Cesaro sum of a series [duplicate]

$\sum_{n=0}^{\infty}a_n$ diverges in the regular term but is Cesaro summable Prove $a_n/n\to 0$ when $n\to \infty$ We used the definition of the Cesaro sum and obtained: $\lim_{N\to ...
1
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1answer
66 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
3
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1answer
41 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
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0answers
29 views

Convergence of orthogonal basis functions

I'm working on a problem where I've generated a set of basis functions using a Laplace series of spherical harmonics to describe the angular part of a 3D distribution and "custom made" basis functions ...
4
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1answer
37 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
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0answers
26 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
1
vote
1answer
37 views

What is the difference between the following $2$ sets?

What is the difference between the following two sets? $\{s\in\mathbb C:\Re(s)\ge1+\delta\},\quad\delta>0$ $\{s\in\mathbb C:\Re(s)>1\}$ I read that $\displaystyle\sum\limits_{n\in\mathbb ...
2
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3answers
73 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
3
votes
1answer
52 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
2
votes
2answers
94 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...