0
votes
0answers
33 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
0
votes
1answer
20 views

Prove this monotone sequence has a bound, thus it converges.

Let $r>0$ and $\frac{r^n}{n!}$ Prove that it converges. I know that it is eventually decreasing, so it is monotone. How do I get a bound for it to show that it converges? Also how would I go ...
0
votes
1answer
29 views

Prove that this sequence converges

I need to show that $ |r^n|$ converges for $0<|r|<1$. I know this converges to $0$. The problem that I have is that I need to use the observation that $\lim_{x\to inf}|r^{n+1}|=\lim_{n\to ...
9
votes
1answer
52 views

Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle ...
1
vote
1answer
25 views

Prove Uniform Convergence of Series of functions-Help?

Let F0 be a bounded Riemann integrable function on [0, 1]. For n ∈ N, define $F_n(x)$ on [0,1] by $F_n(x)$ = $\int_{0}^{x}$ $F_{n-1}(t)$ dt 1) Prove that for all n∈ N and x∈ [0,1], we have ...
0
votes
1answer
22 views

Absolute and conditional convergence of a series with $\sin(x)$

I have to explore absolute and conditional convergence of this function series I tried to find $a(n)$ and $a(n+1)$ terms of the series and then divide it and take a limit. But I've got nothing. ...
1
vote
1answer
35 views

weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
1
vote
0answers
18 views

Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
0
votes
1answer
20 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
0
votes
3answers
48 views

Find a radius of convergence of power series

I have to Find a radius of convergence of this power series I' ve decided to use D'alambert indication: Looking for a limit i meet a problem with a factorial Please. help me finish this ...
1
vote
2answers
136 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? ...
5
votes
0answers
55 views

Forcing series convergence

The following problem was posed by one of my lecturers: $(z_n)$ a null sequence in $\mathbb{C}$. Does there exist $(\epsilon_n)$ with each $\epsilon_n=\pm 1$ such that: $$\sum_n \epsilon_n ...
3
votes
1answer
27 views

Derivative and lipschitz

If I have a real-valued continuous function defined on a compact subset of real line, such that its derivative(wherever it exists) is bounded. Is such a function necessarily Lipschitz? Additionally, ...
2
votes
1answer
18 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
0
votes
0answers
92 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
2
votes
2answers
38 views

Explore the convergence of a series

I have to explore the convergence of a series. At this picture I used radical Cauchy indication. But I don't now what to do with a denominator to find a limit. Help me please ! Thank You so much :) ...
2
votes
1answer
67 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
2
votes
0answers
17 views

Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
1
vote
1answer
22 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
0
votes
1answer
38 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...
0
votes
2answers
32 views

Explore the convergence of a series with ln

How to explore the convergence of this series: $$ \sum_{n=1}^{\infty}\dfrac{1}{\ln^n(n+1)} $$ What would be better to use: De Lamber indication or feature comparison. And if comparison is a good ...
1
vote
0answers
24 views

Comparison of the remainder of a series and its general term

Let $\sum\limits_{n = 0}^{+ \infty} u_{n}$ be a convergent series, such that $\forall n , u_{n} > 0$ My question is : Under which conditions can we find a constant $C > 0$ such that $$\forall ...
2
votes
2answers
55 views

Prove the convergence of $\lim_{n \rightarrow \infty} \frac{23^n}{n^{13}} \cdot \frac{1}{6}$

I have to check for the limit of $$\frac{7^n+23^n-3(11)^n+5^n}{n^2+3n^7+6n^{13}+1}$$ By factorig out $\frac{23^n}{n^{13}}$ and appying the limes to the remaining factor one gets: $$\lim_{n ...
-1
votes
1answer
76 views

real analysis: pointwise limits and uniform convergence [closed]

Find the pointwise limit of the sequence of functions, $\{f_n\}$, on $[0,1]$, where $f_n(x)=2nxe^{-nx^2}$. Use the Extreme Value Theorem to show that the convergence is uniform.
1
vote
2answers
101 views

For which $\alpha$ does the series $\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}}\!\! - \!1\big)$ converge?

For what $\alpha$, does $\displaystyle\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}} - 1\big)$ converge? Divergence of $\sum_{n = 1}^{\infty}(2^\frac{1}{n} - 1)$ prompted this question.
2
votes
1answer
73 views

L1 convergence gives pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are ...
1
vote
3answers
106 views

Multiplication of infinite series

Why multiplication of finite sums $(\sum_{i=0}^n a_i)(\sum_{i=0}^n b_i)=\sum_{i=0}^n (\sum_{j=0}^ia_jb_{i-j})$ (EDIT: This assumption was shown to be false) does not work in infinite case? I have ...
0
votes
2answers
62 views

Is the sequence $(x_n)$ convergent in the space $L_1(0,1)$

Is the sequence $(x_n)$ convergent in the space $L^1(0,1)$ ? $x_n(t)= n^2 t^n (1-t^2)$ for $n\in\mathbb{N}$. norm: $\|x\|=\int_{(0,1)} \left|x(t)\right| \; dt$ I think it should ...
0
votes
1answer
71 views

Find an absotule convergent series that is not convergent

find the sequence of polynomials $(P_n)$ such that $\sum P_n$ converges absolutely (that is $\sum \|P_n\|_{\infty}\lt\infty $) but is not convergent in the space ($\mathcal{P}[0,1], \|.\|_{\infty}$, ...
0
votes
1answer
85 views

Proof of convergence of an infinite product

a) Show that $\Pi_{n=1}^\infty x_n$ converges if and only if for all $\varepsilon>0$ there exists an $N$ such that for all $m\ge n\ge N$, $\left|x_nx_{n+1}\cdots ...
1
vote
0answers
66 views

Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
2
votes
0answers
110 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
1
vote
4answers
70 views

Convergence of an infinite sum

Is it possible to use the comparison test for convergence in the following series? $$\sum_{n=1}^\infty \sin \frac 1 n$$ The exercise says that I should find a linear function $f(x)$ that satisfies ...
2
votes
2answers
173 views

Proof that rational sequence converges to irrational number

Let $a>0$ be a real number and consider the sequence $x_{n+1}=(x_n^2+a)/2x_n$. I have already shown that this sequence is monotonic decreasing and thus convergent, now I have to show that $(\lim ...
1
vote
1answer
23 views

An unknown function

I run into a function: $1_{[-n, n]^r}$. I guess this function equals 1 whenever x falls into $[-n, n]^r$. Am I right? I met this function in an analysis paper which deals with measure and density of ...
0
votes
2answers
40 views

What is the radius of convergence of the power series?

I have the following power series and I would like to figure out the radius of convergence: $$\sum_{n=1}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$ I appreciate any help&explanation. Jacky
1
vote
2answers
31 views

How to determine the Radius of convergence?

I have the function $$ \sum_{n=1}^{\infty} \frac{(-3x)^n}{n^2}$$ I´m not realy sure where to begin and how to determine the radius of convergence. Could someone provide a nice explanation? THX
2
votes
1answer
43 views

Weak Convergence Proof: Tried on My Own

This is related to a question asked here: What a proof of weak convergence is supposed to look like I asked what a proof of weak convergence was "supposed to look like". Specifically, I asked that if ...
0
votes
2answers
27 views

Convergence of $f_{n}(x)=\frac{1}{x^2+n^2}$ and $g_{n}(x)=\frac{2nx}{x^2+n^2}$ in sup norm

I need to show that (i) $f_{n}(x)=\frac{1}{x^2+n^2}$ converges to the zero function in sup norm, and (ii) $g_{n}(x)=\frac{2nx}{x^2+n^2}$ does not. Not sure if this is right but would appreciate ...
1
vote
1answer
64 views

Basic analysis - sequence convergence

I'm taking a course entitled "Concepts in Real Analysis," and I'm feeling pretty dumb at the moment, because this is obviously quite elementary... The example in question shows $\lim_{n\to\infty} ...
1
vote
0answers
40 views

Behaviour of $\sum_{n=1}^{\infty}\frac{z^{n}}{n}$ for $|z| = 1$ [duplicate]

How to show that the power series $$\sum_{n=1}^{\infty}\frac{z^{n}}{n}$$ ( which has radius of convergence $1$ ) converges in all points of $\partial D(0,1)$ except $z = 1$ ?
1
vote
1answer
32 views

Convergence almost everywhere and absolute value

Is it true that if $(f_n)$ converges almost everywhere to $f$ then $(\vert f_n \vert)$ converges also almost everywhere to $\vert f \vert$ ? Thanks.
3
votes
1answer
59 views

Comparison between infinite products and series

I need examples of the following facts: 1) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges $\nRightarrow \prod_{j=0}^{+\infty}(1+|a_{j}|)$ converges 2) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges ...
0
votes
1answer
18 views

Exponent of convergence of $\{m+in \ | \ m, \ n \in \mathbb{Z}, (m,n) \neq (0,0) \}$

The exponent of convergence $\rho$ of a sequence $\{z_{n}\} \subset \mathbb{C}$ is defined as $$\rho = \inf \{\lambda \geq 0 \ | \sum_{n}\frac{1}{|z_{n}|^{\lambda}} < +\infty\}$$ My doubt is : ...
0
votes
1answer
34 views

$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$. Any hint ?
0
votes
2answers
92 views

Prove $x^n$ is not uniformly convergent

This question pertains to the sequence of functions $f_n(x)=x^n$ on the interval $[0,1]$. It can be shown this sequence of functions ${f_n}$ converges point-wise to the limit $f$ where $f$ is defined ...
0
votes
0answers
14 views

Rate of Convergence of matrices

Given a symmetric-"pentadiagonal" matrix $(n\times n)$ such as: $$\begin{bmatrix} 1 ...
0
votes
1answer
56 views

Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \ $, $ z \in \mathbb{C}$

I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$ If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \ $?
1
vote
1answer
57 views

Problem on $L^2$ spaces.

Let $f_n$ be sequence of continuous functions on $[0,1]$ converging uniformly to $f$ a.e. on a set of finite measure. I would like to prove that this implies $f_n\rightarrow f$ in the $L^2$ norm. ...
1
vote
1answer
30 views

Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...