Convergence of sequences and different modes of convergence.

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3answers
82 views

Is there a nicer way to show that the series is convergent?

I'd like to show that for a fixed $z\in\mathbb C\setminus\mathbb Z$ the series $$\sum_{n=1}^\infty \left| \frac{1}{z-n} + \frac{1}{n} \right|$$ is convergent. I think, one can do it as follows. Fix ...
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1answer
22 views

convergence in probability of division and their expected values

Let $\frac{X_n}{Y_n} \rightarrow 1$ in probability. Then does $\frac{\mathbb{E}[X_n]}{\mathbb{E}[Y_n]} \rightarrow 1$? If not, what are the conditions required for this to hold?
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1answer
27 views

Prove the convergence of sequence

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $, with $ f(x) = \sqrt[3]{3x^2 - 2x^3} $ Let $ x_{0} \in (0, 1) $ and $ x_{n + 1} = f(x_{n}), \forall x \in \mathbb{N} $ Prove that $ (x_{n}) $ is ...
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2answers
43 views

Almost sure convergence implies convegence in distribution - proof using monotone convergence

I'm trying to understand the following proof of the statement : "Almost sure convergence implies convegence in distribution" The definition of convergence in distribution is given as follows : $X_n$...
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1answer
75 views

On convergence of particular functions in Sobolev space

I'm having trouble showing the following fact: Suppose $1\le p<\infty$, $\ \varphi\in C^\infty(0,\infty)$ such that $$\varphi(s)= \begin{cases} 0, & s\leq 1/2\\ y\in[0,1],& s\in(1/2,...
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1answer
33 views

Determine the $\alpha$'s for which the integral converges $\int _0^4\:\frac{\sqrt{x}}{\left(4-x\right)^{\alpha }}dx $

I have a problem that I don't know how to solve. I have to determin all $ \alpha $'s for which this integral converges: $$ \int _0^4\:\:\frac{\sqrt{x}}{\left(4-x\right)^{\alpha }}dx $$ What I've tried:...
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1answer
28 views

Limit of a sum where each term is a function of the summation limit

I want to prove that the following limit is finite: $$\lim_{k\rightarrow \infty} \sum_{i=1}^{k-1} \frac{2^i-1}{2^k-2^i}$$ Actually, it would be nice to prove that $$\lim_{k\rightarrow \infty} \sum_{...
2
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1answer
28 views

Convergence of exponential of monotonic function?

Let $f:{\mathbb R}_+\rightarrow{\mathbb R}_+$ be an increasing continuous function. We now that $$\lim_{r\rightarrow \infty} \left(1+\frac{f(x)}{r}\right)^{r} =e^{f(x)}.$$ Then $$\lim_{r\rightarrow \...
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1answer
54 views

Let $f:[0,\infty)→[0,\infty)$ be a continuous function s.t. $\int_{0}^{\infty}f(x)dx<\infty$

Let $f:[0,\infty)→[0,\infty)$ be a continuous function s.t. $\int_{0}^{\infty}f(x)dx<\infty$. Which of the following are necessarily true? $(1)$ The sequence ${f(n)}_{n\in \mathbb N}$ is bounded $...
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1answer
55 views

Normal Convergence in Unit Disk

As I'm preparing for my qualifying exams, I have been given a question, and I'm not sure how to interpret what is being asked. The text I am using is Complex Analysis by Freitag (although the prompt ...
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2answers
39 views

Does $(\vec{a_k})_{k\in \mathbb{N}}$ converge? And if so, what is the limit? [closed]

If $\vec{a}$ is defined like this: $$\vec{a_k}=\binom{\frac{1}{k+1}\cos{k}}{\;\;(k+1)\sin{(\frac{1}{k+1})}\;\;}$$ Does it then converge? And if so, what is the limit? Any help on how to approach this ...
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1answer
50 views

What is the domain for which this integral transform is defined?

Let $s=\sigma+it$ the complex variable where thus $i^2 =-1$, and $\sigma$ and $t$ are real numbers. Let $\mu(k)$ the Möbius function. It is possible determine the set of functions such that $$M \...
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0answers
11 views

A continuity in norm where $p^k\in R^n_{++} $ and $p\in R^n_{+}$

Suppose $X\subset R^n$ is convex and compact and the vector $0\notin X$. Let $p\in R^n_{++}$ and $\langle x,y\rangle_p:=\sum_{i=1}^n p_ix_iy_i$, and $z=\arg\min_{x\in X}\|x\|_p$. Let $$\{x:\...
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4answers
79 views

how to prove $\lim_{x\to \infty}\frac{n}{n^3+1}=0$ using definition of limits?

$$\lim_{n\to \infty}\frac{n}{n^3+1}=0$$ I know that to prove this i have to find a $N \in \Bbb N$ $\forall \epsilon>0$ s.t if $n>N $ then $|\frac{n}{n^3+1}-0|< \epsilon$ i tried it some ...
3
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2answers
50 views

Determine for which value of x the infinite series converges

$$\sum_{n=1}^\infty\frac{\sqrt{\vphantom{A^b}n+1}-\sqrt{n}}{n^x}$$ I multiplied by the conjugate and was able to simplify this to $$\sum_{n=1}^\infty\frac{1}{n^x(\sqrt{\vphantom{A^b}n+1}+\sqrt{n})}$$...
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1answer
28 views

A_n in L^2(R) convergence study

I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $A_nf(x)=\int_\mathbb{R}e^{-n(x-y)^2}f(y)dy$ Solving.... It's easy to see that $A_n\to A=0$ pointwise, now i want to show strong ...
2
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1answer
81 views

About $ \int_{(0,1)^n} \frac{1}{x_1^{\alpha_1}+x_2^{\alpha_2}+…+x_n^{\alpha_n}} dm_n $

I'd like to prove the following two results, but besides the "trivial" implications, I haven't been able to crack them: $$ \int_{(0,1)^n} \frac{1}{x_1^{\alpha_1}+x_2^{\alpha_2}+...+x_n^{\alpha_n}} ...
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0answers
7 views

Convergence of Ritz polynomial in mean square

If i use a method of weighted-residual or Ritz method and obtain a numerical approximation as a polynomial ... How can i prove the convergence of this solution to the exact(in mean square sense)?
2
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2answers
50 views

If $a_n\ge0$ and $\sum a_n$ converges then $\sum\sqrt{a_na_{n-1}}$ converges, what about the converse?

Suppose the series $\sum_{n=1}^{\infty}{a_n}$ is convergent ($a_n \geq0$), Is it true that $\sum_{n=1}^{\infty}\sqrt{a_na_{n-1}}$ is convergent ? Is the converse true? My attempt: The first part I ...
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0answers
47 views

Show that $L^2(0,1)=\operatorname{span}\{e^{2\pi inx}\}_{n\in \mathbb Z}$

Q1) In a course, it's written that $L^2(0,1)$ is spanned by $\{e^{2\pi inx}\}_{n\in\mathbb Z}$. How can I show it ? Q2) Let $f\in L^2(0,1)$. Then we have Parseval equality, i.e. $$\left\|f\right\|^2=...
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4answers
105 views

Prove that if $\left\{ x_{n}\right\} $ converges then $\left\{ \left(x_{n}\right)^{2}\right\} $ converges.

Good night. I have a problem with this problem. I tried the following: Proof: Let $\left\{ x_{n}\right\} $ be a convergent sequence. By definition: $\mid x_{n}-x\mid<\epsilon$ Then: $\mid\...
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1answer
52 views

brownian sample path

I'm currently revising for a probability course and I just came across the following lemma Let $(B_t)_{t\geq 0}$ be a one-dimensional Brownian motion and $0=t_0^n<...<t_{p_n}^n=t$ be a ...
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0answers
52 views

Finding where {$f_n(x)$} converges, where $f_n(x)=x(1-x^n)$

I'd appreciate a double check of my work and logic, and some help answering some questions. 1.) Finding where {$f_n(x)$} converges, $f_n(x)=x(1-x^n)$ Using the ratio test: $$L=lim_{n\to \infty} \...
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1answer
80 views

Convergence of a series of bounded linear operators to zero.

I'm working on a proof and I have a term I can't seem to handle. The problem at hand can be isolated to the following: Let $(T_n)$ be a sequence of bounded linear operators that are uniformly ...
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1answer
33 views

Evaluate convergence radius for $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$

Follow-up question (see "Edit") Given $f(x) :=$ $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$, I have to evaluate the largest open interval where $f(x)$ converges. ...
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1answer
54 views

Why does $\sum_{n=1}^\infty \frac{\cos x^n}{n}$ converge on $-1\le x \lt 1 $, and uniformly converge on $(-1,0)$

Given the series: $$\sum_{n=1}^\infty \frac{\cos x^n}{n}$$ This is similar to $$\sum_{n=1}^{\infty}\frac{x^n}{n}$$ This series converges on $-1 \le x \lt 1$ and converges uniformly on $(-1,0)$ ...
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0answers
69 views

How to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...
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0answers
36 views

Convergence of measures in total variation sense

Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures. Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ ...
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1answer
32 views

We can say that any Cauchy sequence converges to some value in some space?

Maybe this is a trivial question but I need to ask and clarify myself: I know that any Cauchy sequence converges to some value inside some space if this space is complete. But, we can say that any ...
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1answer
23 views

Convergence of $\sum_{n\in\mathbb{N}_{>0}} f(n)\mathrm{log}(n)$

A function $f(n)$ has the following conditions: $$ f(n),n\in\mathbb{N}_{>0} $$ $$ f(n)\in[0,1] $$ $$ \sum_{n\in\mathbb{N}_{>0}} f(n)=1 $$ Does the following sum always converge? Or does a ...
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1answer
45 views

a.s. convergence and conditional expectation

I have a stochastic process $X(t,\omega)$ which is a martingale. It is showed that there exists a r.v. $X(\infty)$ such that in $L^1(\Omega)$, $\lim_{t \rightarrow \infty}X(t) = X(\infty)$. In my ...
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0answers
35 views

Pass to the limit under the sign of integral

I need to show that this integral converges to its limit, showing this only for a subsequence is also good enough for me. Consider $w_k \rightarrow w $ in $L^2$, $u_k \rightarrow u $ weakly in $H^...
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2answers
57 views

Convergence of an improper integral $I=\int_0^\infty \frac{\sin{x}-x\cos{x}}{x^\alpha}dx$

Problem: Analyze convergence of an improper integral $I=\int_0^\infty \frac{\sin{x}-x\cos{x}}{x^\alpha}dx$. My work: Problematic points are $0$ and $\infty$. Therefore, we will write integral as a ...
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3answers
38 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...
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1answer
45 views

Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
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1answer
49 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...
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1answer
38 views

$\lim X_n = 0$ iff $b > 0$

Probability with Martingales: It looks like $$\lim \exp\{aS_n - bn\} = 0$$ if $b > 0$ because $$\lim aS_n - bn = -\infty \tag{*}$$ but how to prove $(*)$?
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5answers
70 views

how to determine if this series converges?

I was trying to find out if the series: $$\sum^{\infty}_{n=1}n^3e^{-n} $$ converges. I tried applying the Cauchy test, $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{n^3e^{-n}}=\...
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1answer
18 views

$((n-1)^{0.5})/(((n+1)^2)-1)$ Is the sum convergent?, why or why not? [closed]

$$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks.
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1answer
22 views

Sufficiently proving a sum converges

This might be a bit basic, but I'm pretty sure I'm wrong about this so I'd appreciate at least a confirmation that I'm wrong. The question is: Prove or disprove: If $\sum_{n=1}^{\infty} ...
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0answers
15 views

Radius of convergence for the following power series using the ratio test

I am slightly unsure about how to do the following question relating to the radius convergence (using specifically the ratio test). The power series is as follows: $$\sum_{n=1}^{\infty}\frac{(2x+1)^n}...
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2answers
100 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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2answers
47 views

For which $ \alpha >0 $ the integral $\int _0^3\:\frac{x}{\left(9-x^2\right)^{\alpha }}dx$ converges?

Good evening to everyone. I have a problem and I don't know where to strat from. I have to find for which $ \alpha >0 $ the integral $$\int _0^3\:\frac{x}{\left(9-x^2\right)^{\alpha }}dx$$ ...
1
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4answers
58 views

For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment?

For $x\in \left ( 0,1 \right )$ converges $\sum_{n=0}^{\infty}\left ( n+2 \right )\left ( x-1 \right )^n$. Is this true statment? To examine convergence I usually use rules for convergence, but what ...
1
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1answer
26 views

Proof radius of convergence for zero and infinite (power series)

I deleted my last question because there was a huge mistake inside. Given: $R$ is the radius of convergence of $\sum_{n=0}^{\infty} a_{n}x^{n}$, also suppose that $\lim_{n\rightarrow \infty} \left | \...
2
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2answers
59 views

How to determine convergence of a rooted parameter?

Ok, so this one stumped me completely. Determine for which values of $\alpha$ the following sum converges: $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{4+n^\alpha}}$$ My gut instinct was that for any $...
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0answers
38 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
2
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3answers
46 views

Uniform convergence of $\sum_{n=1}^{\infty}\frac{x}{n(n+x^2)}$

Prove that $\sum_{n=1}^{\infty}\frac{x}{n(n+x^2)}$ converges uniformly on $[0,\infty)$. On $[1,\infty)$, we have $\frac{1}{n}\leq x$ which implies $\frac{x}{n(n+x^2)}\leq\frac{1}{n^2}$. Since $\sum_{...
0
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0answers
15 views

Vector elements converging to the same value - a proof by contradiction

Note: I'm going to simplify the proposition and proof in this question a bit to avoid a large number of definitions and theorems - hopefully I don't remove anything vital. I'm afraid the material here ...
1
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1answer
67 views

Rearrange the series $ \sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ to converge to $1$.

I have studied the Riemann's theorem about rearrangement of conditionally convergent series. Also I have seen other rearrangements of the given series on this site that converge to different sums $\...