Convergence of sequences and different modes of convergence.

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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 ...
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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 ...
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2answers
47 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 ...
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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?
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3answers
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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
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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: ...
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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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32 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 ...
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38 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 ...
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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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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
36 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 ...
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1answer
63 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 ...
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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. ...
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0answers
28 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 ...
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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 ...
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3answers
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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: ...
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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 ...
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2answers
93 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 ...
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vercongent sequences

Definition- We say a sequence $(x_n)$ verconges to $x$ if there exist an $\epsilon>0$ such that for all $N\in \Bbb{N}$, $n\ge N \implies |x_n-x|<\epsilon$. Loosely speaking, by convergent ...
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91 views

Convergence to $0$ of a certain series.

I was wondering whether or not the following holds - I didn't manage to get anywhere using standard tricks from elementary analysis. ...
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87 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
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1answer
48 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
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Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [closed]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
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0answers
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How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
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56 views

How to evaluate $\lim\limits_{n \to\infty} \left|\frac{(n+1)^n x^{(n+1)}}{n^n}\right|$?

I'm trying to find the radius of convergence for the series $\sum_{n=0}^{\infty} \frac{n^n}{n!}x^n$ and have used Wolfram Alpha to find that it is $|x| < \frac{1}{e}$ and am trying to show that ...
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Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
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74 views

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to ...
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1answer
133 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
3
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1answer
67 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
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1answer
43 views

If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$. [closed]

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
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1answer
43 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
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1answer
79 views

When does interchangibility of limit and Riemann integral imply uniform convergence?

Let $\{f_n\}$ be a sequence of real-valued functions defined on an interval $[a,b]$ such that each $f_n$ is Riemann integrable, $\{f_n\}$ converges point-wise to $f$, $f$ is Riemann integrable and ...
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What is the limit of this product? (SOLVED)

What does this limit equal? $$\lim\limits_{k\to\infty}\left(\prod_{n=1}^kn^{2^{k-n}}\right)^{\frac{1}{2^{k-1}-1}}$$ All that I have tried so far is computation and it does seem to converge. I ...
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26 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
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1answer
34 views

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$? Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of ...
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6answers
81 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
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1answer
39 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
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28 views

A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
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1answer
39 views

Simple Question about Monotone Convergence Theorem

Suppose we have a sequence of (discrete) random variables $X_0, X_1, \dotsc$ over $E$ and $A \subseteq E$. Let $Y$ be some other random variable. Moreover, let $Z$ be a random variable with values in ...
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14 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
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4answers
51 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
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44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
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1answer
28 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
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2answers
38 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
2
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1answer
23 views

If $\sum_{k=0}^{r-1} c_k =0 $, and $a_n \to 0$, does $\sum_{n=0}^{\infty} \sum_{k=0}^{r-1} c_ka_{nr+k} $ converge?

This is a generalization of the alternating series convergence result and this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my question: If ...