Convergence of sequences and different modes of convergence.

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3
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3answers
49 views

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. I tried few things but it wouldn't work out. I would appreciate your help.
-1
votes
0answers
11 views

Convergent in the direction [on hold]

Let {x^n}∈R^n converging to x is said to converge in the direction y∈R^n if there is a secuence of positive numbers in -> 0 and limn→∞(x^n-x)/in=y.
0
votes
1answer
16 views

Interpretation of the radius of convergence

What interpretation should one give to the radius of convergence of a series $\sum a_nz^n$ ? I do know how it is mathematically defined and what it implies for convergence/divergence, but I'm having ...
2
votes
1answer
26 views

On existence of a convergent subsequence

Let $(a_{(m,n)})_{m,n \in \mathbb{N}}$ be a double sequence of positive numbers. Suppose that we know that there exists a limit $\lim_{m \to \infty}\lim_{n \to \infty}a_{(m,n)}=L$. Does there always ...
2
votes
2answers
37 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
1
vote
2answers
52 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...
1
vote
1answer
23 views

argument technique to prove convergence of random variable

I witness a lemma in my class note and I think the proof is not quite clear. Could anybody give me some ideas about argument technique to prove the lemma? The lemma 3 in the beginning of the text: ...
1
vote
1answer
23 views

Two series to be tested for convergence.

Investigate the convergence of the two series $$\sum_{n=0}^\infty \frac{(3n)!}{n^{3n}} \\ \sum_{n=0}^\infty (-1)^n \frac{\ln n}{n}$$ Hi, I tried use near every criterion to solve it, but without ...
2
votes
2answers
34 views

Uniform convergence to 0

Let $(f_n)_\mathbb{N}$ be a sequence of continuous functions $[0,1]\to\mathbb{R}$ converging to $0$. The functions are such that for all $x$, $(f_n(x))_\mathbb{N}$ is decreasing. How can one show ...
2
votes
0answers
20 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
1
vote
2answers
16 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
0
votes
0answers
16 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
0
votes
0answers
21 views

If $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$ then $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. [duplicate]

Let $a_n>0$ and let $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$. Prove $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. I am confused by this sort of sequence\sum thing. How can I use ...
0
votes
2answers
25 views

Determine if the series diverge or converge?

So I was wondering what is the best TEST(divergent test, alternating series,power series,ratio test or root test) that we can use for the following series :
2
votes
1answer
46 views

$\frac{S_n}{n} \to -1 \ \ a.e.$, exercise from probability book

I'm stuck with this exercise from Williams, probability with martingales. Let $X_1, X_2, \ldots $be independent random variables with $$P(X_n = n^2-1 )= \frac{1}{n^2}$$ $$P(X_n = -1 )= ...
1
vote
1answer
22 views

Does the mean integral over B(x,r) of a L1 function u converge a.e. to u(x)?

Suppose $u\in L^1(\Omega )$. Let $u_{x,r}$ be the mean of $u$ over the ball $B(x,r)$ (s.t. $B(x,r) \subset \Omega$), i.e. $ u_{x,r} := \frac{1}{|B(x,r)|} \int_{B(x,r)} u(y) dy$. Is it true that ...
0
votes
0answers
32 views

Is there a way to reverse the ratio test?

My question arises from the following problem: Let $ a_n $ be a real series, so that $ \sum_{n=1}^ \infty a_n $ converges and $a_n \ge 0 $ and $a_n$ monotonously decreasing. It is to prove: $ ...
0
votes
2answers
41 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
1
vote
1answer
20 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
2
votes
1answer
38 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
0
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0answers
9 views

On Differentiation Theorem and Radius of Convergence

For reference: http://www0.maths.ox.ac.uk/system/files/coursematerial/2014/2644/48/14MT-AnalysisI-sheet7.pdf 4.(b) For fixed $d \in \mathbb{R}$, let $f_d(x)=S(d+x)C(d-x)+S(d-x)C(d+x)$ By ...
2
votes
1answer
38 views

Convergence in measure - product

I'm trying to prove the following statements in Folland's book. Let $(X,\mathcal{M},\mu)$ be a measure space. If $f_n\to f$ in measure and $g_n\to g$ in measure, then $f_n+g_n\to f+g$ in measure and ...
0
votes
1answer
28 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
0
votes
3answers
105 views

convergence of $(1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$

I was given a problem of testing sequence convergence. The sequence is defined as: $$x_n= (1+\frac{1}{2})(1+\frac{1}{4})\ldots(1+\frac{1}{2^n})$$ My first idea was to define $y_n$ as follows: $$y_n = ...
1
vote
1answer
26 views

Please check my answers on these Convergence Problems

Let $X_1,X_2,...$ be a sequence of random variables with corresponding distribution functions given by $F_n(x)=0$ if $x<-n$, $F_n(x)=\dfrac{x+n}{2n}$ if $-n\leq x< n$ and $F_n(x)=1$ if $x\geq ...
1
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2answers
61 views

Question about series convergence $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$

So I have been playing around with convergent series recently and I still have a hard time understanding why $\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges. ...
1
vote
1answer
17 views

Radius of convergence: Why is it $\geq 1$?

Let $X$ denote a random variable with values in $\mathbb{N}_0\cup\left\{\infty\right\}$. Let $r_X$ denote the radius of convergence of $\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)z^n$ with ...
1
vote
2answers
50 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
1
vote
0answers
24 views

Convergence of a recursive sequence of functions

Consider the sequence of functions $f_n:[0,1]\rightarrow[0,1]$ defined recursively: $$f_n(p)= 1-p + p (f_{n-1}(p))^2 \quad f_0(p)=1-p \quad f_n(1)=0$$ Computationally one can check that $\{f_n(p)\}$ ...
0
votes
0answers
21 views

Cauchy sequences and convergent sequences?

I am wondering if one can assume that any Cauchy sequence in $(X,d)$ converges to some point in some larger space $(\hat X,d)$ with the same metric. Take for example $(\Bbb Q,d)$. It is not complete ...
1
vote
1answer
37 views

An Elementary Convergence Problem in Probability

Suppose that $X_1,X_2,...$ are degenerate random variables such that $f_{X_n}$ denotes the mass function of $X_n$.$$f_{X_n}(x)=P(X_n=x)= \begin{cases} 1, & x=2+\dfrac{1}{n} \\ 0, ...
1
vote
1answer
13 views

Supremum of cadlag functions

Let $f_n,f$, $n\in\mathbb{N}$, be (real-valued) cadlag functions on $[0,1]$ such that $$\sup_{0\le t\le 1}|f_n(t)-f(t)|\to 0\text{ as }n\to\infty.$$ Does someone have an idea how to prove that ...
0
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0answers
19 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
0
votes
1answer
9 views

convergence at a specific value

I have a series $S_n=\sum_{i=1}^{n} a*(1-i)$ where $a$ is an unknown constant independent of $i$. Is there a way to figure out for which $n$ the above expression converges to the value 0.01? After ...
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0answers
20 views

Convergence in distribution plus convergence of moments.

Suppose that the sequence of r.v $\{X_n\}_{n\geq 1}$ has all the moments, and $X_n\stackrel{D}{\longrightarrow} X\sim N(0,\sigma)$. Assume that $E\{(X_n)^K\} \stackrel{n} {\longrightarrow} E(X^K)$, ...
3
votes
4answers
86 views

$\sum_1^n 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} $ converge or not? [duplicate]

how to check if this converge? $$\sum_{n=1}^\infty a_n$$ $$a_n = 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1}$$ what i did is to show that: $$a_n =2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} > 2\sqrt{n} - ...
3
votes
1answer
179 views

How to compute this kind of limit? [duplicate]

Let $x_0=a,x_1=b$ $$x_{n+1}=\Big(1-\dfrac{1}{2n}\Big)x_n+\dfrac{ x_{n-1}}{2n}, n\ge1$$ Find $\lim x_n.$ If limit exist I can plug limit as $l$ to get a equation of $l$, whose root will be the ...
2
votes
2answers
31 views

A sequence of Continuous Functions Converges Uniformly over $\mathbb{R}$ if it Converges Uniformly over $\mathbb{Q}$

I'm trying to show that if ${f_n}$ is a sequence of real functions that is continuous over all of $\mathbb{R}$ and that converges uniformly to $f$ over $\mathbb{Q}$, then it converges uniformly to $f$ ...
2
votes
2answers
55 views

$x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then lim $x_n$=? [closed]

Let $x_0=a,x_1=b$,if $x_{n+2}=\frac 13 (x_n+2x_{n+1})$, then $\lim x_n=?$ I see here a geometrical interpretation of the points of sequences. It is $ x_2$ the inner point 2:1 of a and b. ...
2
votes
1answer
34 views

Limit of products in $L^p(\mathbb R^d)$

Fix $1 \leq p < \infty$. If $f_n \to f$ in $L^p(\mathbb R^d)$, $g_n \to g$ pointwise, and $\| g_n \|_{\infty} \leq M < \infty$ for all $n$, prove that $f_ng_n \to fg$ in $L^p(\mathbb{R}^d)$. ...
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0answers
22 views

What can be said about the limit of a converging infinite polynomial?

Suppose we have the following polynomial of infinite order: $f(x) = a_0+a_1x+a_2x^2+...=\sum_{n=0}^{\infty}a_nx^n$ Also suppose that $f(x)$ converges to some limit $f^*$ as $x\nearrow\infty$, i.e. ...
0
votes
3answers
50 views

Proving the reciprocal of a divergent sequence is convergent

I want to prove that given a sequence (an) where the limit as n --> infinity of (an) = infinity, the limit as n --> infinity of 1/(an) = 0. It's introductory real analysis, but I'm not sure where to ...
0
votes
1answer
53 views

Convergence of an integral $\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$

For what $\alpha\in\Bbb{R}$ does $\displaystyle\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$ converge ? The $0$ bound doesn't seem to be much of a problem, but I don't see how to deal with the ...
8
votes
0answers
217 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
-1
votes
1answer
12 views

does function converge uniformly

Does the sequence $(f_n)$ of functions $f_n:(0,\infty)\to\mathbb R$ $$f_n(x) = \begin{cases} 0 & x \in (0,\frac1{n+1}) \\ \sin^2 \frac\pi x& x \in [\frac1{n+1},\frac1n]\\ 0 ...
1
vote
0answers
35 views

Test $\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$ for convergence with asymptotic comparison

Suppose we want to test the following series for convergence $$\sum_{n=1}^{\infty}\Big(1-n \sin(\frac{1}{n})\Big)$$ I have found a solution that uses the asymptotic comparison test and uses the ...
2
votes
1answer
76 views

Is this series: $\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}}$ convergent?

How can I show that the following series is convergent or divergent ? $$\sum_{n=1}^{\infty}{{1 \over n} \cos{(n)} \sin{(nx)}},x\in \mathbb{R}$$ I want to use Abel-Dirichlet criteria. I've ...
3
votes
5answers
99 views

How to prove this limit: $\lim\limits_{n \to \infty} \frac1{\sqrt[n] n}=1 $? [duplicate]

This is what I am trying to prove: $$\lim_{n \rightarrow \infty} \frac1{\sqrt[n] n}=1 $$ How do I go about this?
1
vote
2answers
79 views

Determine if the following series is convergent or divergent $\sum_{n=1}^{\infty}\frac{ \sin{nx}}{n}$

I've been doing exercises with series for some time and now I've got an exercise that I'm supposed to solve using Abel or Dirichlet criteria. The problem is : Determine if the following series ...
-2
votes
1answer
52 views

Does $\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}$ converge?

Does $$\int_0^1 \sqrt{\frac{1+x}{\sin{x}}}dx$$ converge? I have tried to substitiute $x$ in nominator as $\tan{x}$ and simlify it using trigonometric formulas, but the integral was still too ...