Convergence of sequences and different modes of convergence.

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Math Analysis - Problem with showing sequence of functions is convergent and uniformly convergent

Let $f:\left[\frac{1}{2} ,1\right] \rightarrow \mathbb R$ be a continuous function, $\{g_n\}_{n=1}^{\infty}$ a sequence of functions where $g_n(x) = x^n f(x)$, with $x \in \left[\frac{1}{2} ,1\right]$ ...
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242 views

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on ...
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2answers
132 views

Radius of convergence of a general series

If the radius of convergence of $\sum a_n x^n$ is $R$, prove that the radius of convergence of $\sum a_n x^{2n}$ is $R^{1/2}$. What I tried to do is look at $$\sum \sqrt{a_n} x^n $$ Then we get that ...
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1answer
144 views

A Coupled Random Walk on the xy-Plane

Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows: ...
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2answers
165 views

Extracting a subsequence from a sequence of $\mathcal{L}^1$ functions

Any help with the following problem is appreciated. Given: a sequence of nonnegative functions $(g_n)$ which are U.I. (uniformly integrable) in $\mathcal{L}^1(0,1)$ with $\sup_n \Vert g_n \Vert_1 ...
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If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.

A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$ Let $(X,d)$ be a metric space and let ...
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1answer
41 views

Baire one functions, closed intervals

I've been wondering if you could help me with the following problem. There's an article on Baire one functions (number 2 on google search list) and there is one thing concerning Lemma 9 that I am not ...
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1answer
33 views

Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?

For example, suppose $$ x_n = \frac 1n \\ X = (0, 1)$$ Is $x_n$ convergent in $X$? My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
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Convergence of a sequence $c_n$

Suppose that $(a_n)$ and $(b_n)$ be sequences such that $\lim (a_n)=0$ and $\displaystyle \lim \left( \sum_{i=1}^n b_i \right)$ exists. Define $c_n = a_1 b_n + a_2 b_{n-1} + \dots + a_n b_1$. Prove ...
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1answer
809 views

binomial expansion formula proof, bases on Lagrange form of Taylor series remainder

Another exercise from Bartle/Sherbert Introduction to Real Analysis book (this one is exercise 9.4.14): Use the Lagrange form of the remainder to justify the general Binomial Expansion ...
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1answer
226 views

Does a convergent power series on a closed disk always converge uniformly?

If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
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1answer
64 views

Baire one functions, characteristic functions of intervals

Do you think you could help me prove that characteristic functions of intervals are Baire one functions? And is it true that linear combinations of Baire one functions are also Baire one?
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1answer
144 views

Laplace Transform of a Brownian motion

If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
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1answer
203 views

Pointwise limits of continuous functions

Could you help me prove the following? Let S be the set of function that are the pointwise limit of continuous functions, $\{h _n\} \subset S$ with max$_{x \in [0,1]} |h_n(x)|< A_n$ and $\sum ...
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0answers
55 views

Is this pointwise convergence sequence also uniform convergence?

$f_{n}$ and $f$ are continuous functions and $f_{n}\rightarrow f$ pointwise. Which of the following are correct? $\int _{0}^{x}F_{n}\left( t\right) dt\rightarrow\int _{0}^{x}F\left( t\right) dt$ ...
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1answer
73 views

Radius of Convergence of $\sum a_nx^n$

Not really sure what I'm missing on this problem: Find the radius of convergence for the following: $$\sum a_n x^n= \sum \frac{(3n)!}{(n!)^2}x^n$$ From my understanding: ...
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1answer
87 views

convergence of complex series.

For what values ​​of $(a,b\in \mathbb{C})$ does this series converge or diverge? $\sum\frac{(k-a)^2}{(k-b)^3}$ if $a,b\in\mathbb{R}$ by the Limit comparison test (with $\sum\frac{1}{k}$ ) I know ...
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4answers
212 views

Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$

How can I prove that for every $ x \notin \mathbb Z$ the series $$ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$$ converges uniformly in a neighborhood of $ x $?
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0answers
194 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
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1answer
84 views

Isn't $f_n(x)=\frac{{n^2}\ln x}{x^n}$ with $x\geq1$ uniformly convergent by $T$-test?

One Dr. showed to me that the function $f_n(x)=\dfrac{{n^2}\ln x}{x^n}$, $x\geq1$ is not uniformly convergent by $T$-test, but I showed it to converge to $0$ anyway. $$\lim_{x\to \infty}T_n=\lim_{x\to ...
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242 views

Kolmogorov's maximal inequality and convergence of random series.

Let $(X_n)_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S_n = \sum_{l=1}^n X_l$. Prove that for any ...
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1answer
115 views

Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$

I found an assertion in this paper at the beginning of page 6, but i can't see how to justify it: Let $X_n \geq 0$ i.i.d. with finite expectation then: $$ \frac1n\max\limits_{k \leq n}X_k \to 0 ...
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1answer
131 views

If $x_{m+n} \le x_n+x_m$, then $\lim x_n/n$ exists and is equal to $\inf x_n/n$

Let $(x_n)_{n \ge 1}$ be a sequence of real numbers satisfying $$x_{m+n} \le x_n+x_m$$ $m,n \ge 1$. Show that $\lim \limits_{n \to \infty} \dfrac{x_n}{n}$ exists and is equal to $\inf \left ...
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1answer
184 views

Point convergence, uniform convergence and near uniform convergence of infinite series $ f_n = x^2 e^{-nx}$

Please help me in prove / decline the point convergence, uniform convergence and near uniform convergence (comapact uniform convergence) of $\sum_{n=1}^{\infty} f_n$ where $f_n : [0, +\infty) ...
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2answers
46 views

Define $f: I \rightarrow \mathbb{R}$ as $f(x)= \sup {f_n(x) : n \geq n_0 }$ for $ x \in I$, It's convex?

Suppose, that $f_n:I\rightarrow \mathbb{R}$ are convex functions for $n\geq n_0$ and $\forall_{x\in I} \exists_{y\in \mathbb{R}} \forall_{n\geq n_0} f_n(x)\leq y$ Define $f: I \rightarrow ...
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1answer
27 views

$\sup\{t_{1}f_{n}(x_{1})+t_{2}f_{n}(x_{2})\mid n\geq n_{0}\}=\sup\{t_{1}f_{n}(x_{1})\mid n\geq n_{0}\}+\sup\{t_{2}f_{n}(x_{2})\mid n\geq n_{0}\} $

I was thinking, if this is correct: Let $f_n$ is a series of convex, limited function $I \rightarrow \mathbb{R}$ $t_1, t_2 \in \mathbb{R} \ \ \ \ \ t_1 + t_2 = 1$ Is that a true statement : ...
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3answers
1k views

$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$

Please help me check, if $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$ $f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
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1answer
78 views

Behavior at infinity.

Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$ So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I ...
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1answer
297 views

Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$

What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$? Here's a plot ...
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5answers
115 views

If $\{x_n\}$ is a sequence in an open interval and $f$ is continuous, must $\{f(x_n)\}$ have a convergent subsequence? What if the interval is closed?

I seek to show whether the following statements are true or false: If $f$ is continuous on $(a,b)$ and $\{x_n\}$ is a sequence in $(a,b)$, then the sequence $\{f(x_n)\}$ has a convergent ...
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205 views

Characteristic functions of intervals

Could you explain to me what characteristic functions of intervals are? I'm reading something about convergence of sequences of functions and it says there that characteristic functions of intervals ...
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1answer
131 views

Absolute convergence of $\sum\limits_{n=0}^{\infty} \frac {z} {(z+n)^2}$

I want to check the absolute convergence of $\displaystyle\sum_{n=0}^{\infty} \frac {z} {(z+n)^2}$ in the half plane $\Re(z)>0$, and to see if the convergence is uniform or locally uniform. How do ...
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82 views

Proof of convergence

Prove if the following series converges: 1) $\sum_{k=2}^\infty \frac{k^2}{-1+k^5}$ $$\lim\limits_{n \to \infty}\frac{k^2}{-1+k^5} \cdot\frac{1}{1/k^3}$$ $$\Leftrightarrow \lim\limits_{n \to ...
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2answers
197 views

Intuition for convergence iterative formula

A convergence iterative formula , $g(x)$ , holding that $|g'(z)|<1$ . In a case which the equation is given and I have to evaluate iterative formula in order to find its fixed point . For ...
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3answers
77 views

convergence of $\sum\frac{a_{n}}{n}$ if $\sum_{k=1}^{n}a_{k}\le M*n^{r}$ where $r<1$

Show that if the partial sums $s_{n}$ of the series $\sum_{k=1}^{\infty}a_{k}$ satisfy $|s_{n}|\le M*n^{r}$ for some $r<1$, then the series $\sum_{n=1}^{\infty}\frac{a_{n}}{n}$ converges.
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Radius of convergence in a series. Ratio test.

I am having a hard time with this question. $$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$ I used the ratio test and got stuck here: $$x^2 \lim_{k\to\infty} \frac ...
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1answer
180 views

Does $\sum _{n=1}^{\infty}\frac{1}{n} ((-1)^n+\frac{1}{n})$ converges?

$$c_n=\sum _{n=1}^{ \infty}\frac{1}{n} ((-1)^n+\frac{1}{n}) =\sum _{n=1}^{ \infty}\frac{1}{n} (-1)^n +\sum _{k=1}^{\infty} \frac{1}{n^2}$$ $a_n=\sum _{n=1}^{ \infty}\frac{1}{n} (-1)^n$ $b_n=\sum ...
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Time steps and convergence

I was wondering if someone could please give me an explanation why we look at larger and larger time steps in order to see if we have convergence. I have played around with matlab and noticed when I ...
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0answers
108 views

Monotonic convergence of powers of a stochastic matrix

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution ...
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1answer
87 views

Convergent & Cauchy Sequence related prove

(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $\{a_n\}\to a$ and $\{b_n\}\to b$. Prove that $\{a_n+b_n\}\to a + b$ (2) Prove that a convergent sequence is Cauchy. ...
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298 views

Proof that $\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2}$ convergences uniformly using the Cauchy criterion

I'd like to use the Cauchy criterion to show that $$f_n(x)=\sum\limits_{k=1}^n\frac{\sin(kx)}{k^2} \mbox{convergences uniformly} $$ Here is what I did: We want to show that $\forall \, ...
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1answer
72 views

Absolute and Conditional Convergence in Spivak

This is question 5 Chapter 23 from Spivak: I have already proved a) which said Prove that if $\sum a_n$ converges absolutely, then so does $\sum (a_n)^3$ and part b said show that this is not true ...
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76 views

Convergence stronger than uniform convergence

Is there any kind of convergence stronger than uniform convergence? As for example $f_n(x)\underset{n\rightarrow+\infty}{\longrightarrow}f(x)$ with a 'form' of convergence that implies uniform ...
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39 views

Why is $|g'(z)| < 1 $ a condition for convergence?

For example this function (its zero is $z \approx 0.5$) - $$f(x) = \displaystyle x+\ln{x} $$ and the iterative formula - $$ \displaystyle x_{n+1} = -\ln{x_n} $$ So iterative formula is $g(x) ...
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1answer
86 views

Dominated convergence theorem

Let $(X_t)_{t\leq1}$ and $(Y_t)_{t\leq1}$ be such that $0\leq X_t\leq Y_t$ w.p. $1$ for each $t$, and assume the following: \begin{equation*} \lim_{t\to0}\mathbb{E}[Y_t]=K \end{equation*} Does that ...
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1answer
180 views

Radius of convergence of $\sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$

I want to find out the radius of the following power series of a complex variable: $\sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$ The ration test $\lim_{n \to ...
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1answer
110 views

Showing that if $|a_n|<|b_n|$ and $\sum b_k x^k$ converges, then $\sum a_k x^k$ converges

Hypothesis: $|a_{n}|<|b_{n}|$ for all natural $n$, and $\displaystyle\sum_{k=0}^{\infty} b_{k}x^k$ converges on $(-R, R)$. Prove that $\displaystyle\sum_{k=0}^{\infty} a_{k} x^k$ converges on ...
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1answer
52 views

$X_n \overset{a.s.}{\longrightarrow} X$ and $X_n \overset{L^1}{\longrightarrow} Y$ implies $X = Y$ a.s.?

If I have a sequence of random variables $\{X_n\}_{n \geq 0}$ such that $$X_n \overset{a.s.}{\longrightarrow} X \quad\textrm{and}\quad X_n \overset{L^1}{\longrightarrow} Y$$ then is it always true ...
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1answer
61 views

Theorem proof: difficulties with one argument

In this pdf, theorem $2.2$ (page $6$ or $103$- since it's not really the whole book) there is a point that I don't really understand fully. $|a_n| < |l| + 1$ for all $n \ge N$, and then the next ...
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1answer
118 views

find the radius of convergence of the following power series

$\sum_{n=0}^{\infty}z^{n!}$ My textbook says its answer is 1. But I think the sequence is not power series because it cannot be expressed as the form of $CnZ^n$. Am I right? Or where do I have a ...