Convergence of sequences and different modes of convergence.

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2
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1answer
50 views

conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$

prove that the series $$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$ is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that $\sum_{n=2}^{\infty} \...
0
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0answers
22 views

Using ratio test for sequences?

Don't mark this as duplicate. The question is to verify whether my method is correct. Prove: $$\lim_{n\to\infty} \frac{x^n}{n!} = 0$$ Method: Let $\sum_{n=1}^{\infty} \frac{x^n}{n!}$. By ratio ...
6
votes
2answers
69 views

Why does Slutsky's Theorem Fail to Generalize? [on hold]

What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that $...
1
vote
1answer
23 views

Convergence of minimum of a sequence of functions

Consider functions $f_n, f$ from $\mathbb{R}$ to $\mathbb{R}$. Suppose $f_n(y)$ converges pointwise to $f(y)$ for all $y$ as $n \rightarrow \infty$. I would like to know under what conditions is the ...
0
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2answers
57 views

Convergence of integral $\int_{0.5}^{1} \log(|\log(t)|)\ ^ 7\,\mathrm{d}t $

Does this integral converge ? $$\int_{0.5}^{1} \log(|\log(t)|)\ ^ 7 \,\mathrm{d}t $$ I have tried to compare this integral to $$\int_{0.5}^{1} \dfrac{1 }{\sqrt{x-1}} \,\mathrm{d}t $$ but couldn't ...
0
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2answers
66 views

Theorem 3.16. in Analytic Number Theory by Apostol

The below texts are from the book Introduction to Analytic Number Theory by Apostol: I have two questions which I couldn't find solutions for them: $1-$ According to Thm 3.16., $\sum_{n\le x} \...
4
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1answer
70 views

Convergence of the integral $\int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx$ as $N\to\infty$

How can calculate this integral $$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$ where $f(x)$ is a probability density function?...
0
votes
1answer
19 views

Random subseries of harmonic series expected to converge, but how often does it?

Inspired by a previous question which I can't seem to find, what if we have $$X = \sum_{k=1}^{\infty}\frac{1}{k}\cdot P\left(U(0,1)<\frac{1}{k}\right)$$ That is, each term of the series will be $\...
0
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0answers
19 views

Summary of results concerning interchange of limits in series

The document http://www2.iugaza.edu.ps/ar/periodical/articles/volume%2014-%20Issue%201%20-studies%20-16.pdf constructs the theory of double sequences and double series very nicely, supplying the ...
2
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0answers
29 views

Confusion about the convergence of Riemann zeta function in terms of the integral

Titchmarsh wrote that $$\zeta(s)=s\int_{1}^{\infty}\frac{\left \lfloor x \right \rfloor-x+\frac{1}{2}}{x^{s+1}}\,\mathrm{d}x+\frac{1}{s-1}+\frac{1}{2}\tag{2.14}$$ using the Euler-Maclaurin summation, ...
7
votes
4answers
103 views

Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
0
votes
2answers
90 views

Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} $$
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0answers
37 views

Converging savings

Let's imagine abstract situation that there is a city where all families have houses (their number is = $1000$) which are located on the edge of a big circle so every house has exactly two neighbors ...
0
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0answers
20 views

Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
1
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1answer
23 views

What condition on the coefficients $a_n$ will guarantee $f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ is k times differentiable?

$f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ What condition on the coefficients $a_n$ will guarantee $f$ is $k$ times differentiable? I'm not sure where to begin with this, because it ...
-1
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2answers
48 views

Determine whether the following integral is convergent or divergent $\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$

Please, help me to determine the following integral: $$\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$$ As we know, this is the II order indeterminant integral. I've tried to use comparison test, ...
3
votes
1answer
51 views

Does iterating $x \cdot \sin(\frac 1 x) + x$ near $0$ approach $0$?

Let $f^1(x) := x\,\sin(\frac{1}{x})+x$ and define $f^N (x):= f(f^{N-1}(x))$ for $N\in \mathbb{Z},\ N>1$. For which $x \in \mathbb{R}$ does $\lim_{N\rightarrow\infty}{f^N(x)}=0?$ Clearly, for $x\...
1
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1answer
65 views

Summing power series $\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$

Lets have series $$\sum_{n=1}^\infty \frac{(2n-1)!!}{(2n)!!}x^n$$ Obviously, its convergence radius is 1. I should sum it, but don't know what's up with the double factorial. There is a hint in the ...
1
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1answer
49 views

Does the series $\frac{1}{(\log{n})^4}$ converge?

I have used comparison test to show it diverges: $$\frac{1}{n}<\frac{1}{(\log{n})^4}$$ But is this even right?
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2answers
36 views

Uniform convergence of following series [closed]

Prove that $\sum_{n=1}^\infty \frac{x^{2n}}{(1 + x + \dots + x^{2n})^2}$ converges uniformly when $x \geq 0$.
2
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2answers
48 views

Testing convergence of series $\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$

Lets have this problem. $$\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$$ I have rewritten this to a form $$\sum\frac{1}{np'^{\ln\ln(n)}q'^{\ln\ln\ln(n)}}$$ For $p,q\in\mathbb{R}$. Obviously, $...
1
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1answer
25 views

Do these series vanish asymptotically?

Let's consider a monotone increasing sequence $(K_N) \subseteq \mathbb{N}$ with $(K_N) \xrightarrow[N]{} \infty$ and $(K_N) = \mathrm o(N)$ (less increasing than $(N)$). Question: $\sum_{j=K_N + 1}^...
0
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2answers
32 views

How do I find the radius of convergence for $\sum_{n=0}^{\infty}\frac{1}{\sqrt{n}}z^n$?

I'm a little unsure about methods on finding the radius of convergence of a function. It would be great to get some help on how to approach these kinds of problems.
0
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0answers
15 views

$Y_n\xrightarrow{P}E(X_i)$

I have this problem at hand $X_1,X_2,\cdots $ are iid random variables with finite second moments.Define$$Y_n={2\over n(n+1)}\sum_{i=1}^n iX_i$$ Show that $Y_n\xrightarrow{P}E(X_1)$. I know ...
1
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1answer
39 views

How to find Taylor series when $x_0=0$ and radius of convergence for $\frac{x}{1+x}$ for $f:(-1,\infty)$

Through the taylor series formula: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n$$ I've got that $f(x)=x-x^2+x^3-x^3\dots$ however my teacher claimed ...
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2answers
40 views

finding the radius of convergence of $\sum_{n=1}^{\infty} n^2x^n$ [closed]

How does one find the radius of convergence of: $\sum_{n=1}^{\infty} n^2x^n$ using the fact that it's possible to differentiate every term. I have no idea how to go about with this
3
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0answers
57 views

Divergence of $\sum \limits_{n=1}^{\infty}\frac{a_n}{a_1+a_2+\dots+a_n}$ [duplicate]

Suppose that $\sum \limits_{n=1}^{\infty}a_n$ series with positive terms which diverges then series $\sum \limits_{n=1}^{\infty}\dfrac{a_n}{a_1+a_2+\dots+a_n}$ also diverges. Can anyone show how to ...
1
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0answers
33 views

Geometric mean of random geometric variables converging with probability to a constant

I have the following question at hand.. Let $X_1,X_2,\cdots, X_n$ be a sequence of iid random variables with common uniform distribution on $[0,1]$. Define $$Z_n=\left(\prod_{\ i=1}^{\ n}X_i \...
2
votes
5answers
111 views

If a limit is finite does it have to be of the form $0/0$?

In my text book it is written that if $$\lim_{x\to0}\;\frac{\cos(4x) + a\cos(2x) + b}{x^4}$$ is finite then $\frac{\cos(4x) + a\cos(2x) + b}{x^4}$ should be of the form $0/0$ and therefore $\cos4x + ...
3
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1answer
59 views

Convergence of $\sum_{n=0}^\infty (\sin{\frac{1}{n}})(\sinh{\frac{1}{n}})(\cos n)$ [duplicate]

$|(\sin{\frac{1}{n}})(\sinh{\frac{1}{n}})(\cos n)|\le|(\sin(\frac{1}{n}))(\sinh{\frac{1}{n}})|$ Since $\lim_{x\rightarrow 0} \frac{\sin(x)}{x}=\lim_{x\rightarrow 0} \frac{\sinh(x)}{x}=1$ Thus $$\...
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2answers
64 views

Is this series divergent or convergent?

Please explain what method you used to prove so. $$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$
0
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0answers
71 views

Does this sequence of random variables converge almost surely?

I was trying to understand why almost sure convergence doesn't imply convergence of the mean and I encountered this answer. However, I do not understand why this sequence of random variables ...
0
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1answer
14 views

Prove/disprove radius about radius of convergnce

I have the following statement - The Taylor series of $\frac{x}{x+2}$ around $X = 1$ has a radius of convergence of $R = 4$. Is it right to say that this statement is false because a function is ...
2
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1answer
20 views

Prob. 4 (b), Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Which of these sequences are convergent w.r.t. the product, uniform, and box topologies?

Let $\mathbb{R}^\omega$ denote the set of all the (infinite) sequences of real numbers. Then which of the following sequences in $\mathbb{R}^\omega$ are convergent (and if so, then to which points(s)) ...
2
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2answers
32 views

Pointwise limit of the sequence of functions $h_n(x)=1,\, \forall x\ge 1/n$ and $h_n(x)=nx,\,\forall x\in[0,1/n)$

Pointwise limit of the sequence of functions $$h_n(x)=\begin{cases}1,&\text{if }x\ge \frac1n\\nx,&\text{if }x\in[0,\frac1n)\end{cases}$$ The trouble with this question is that I think that $...
5
votes
1answer
127 views

Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?

$$\sum_{n=2}^\infty \frac{1}{\ln(n!)}$$ I tried by comparing it to $\sum_{n=1}^\infty \frac{1}{n}$ but i seem to fail. I think I need to compare with series that are smaller and diverge. Help.
3
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1answer
73 views

Weak convergence with uniformly convergent functions.

Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that $$ \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\...
2
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1answer
52 views

Can every element in the arbitrary space be converged to?

When I have for example $\mathbb R$ then I'm able to create a sequence which will converge to any of the elements in $\mathbb R$: \begin{align} \frac{1}{n} &\rightarrow 0\\ \frac{1}{n} + 1 &\...
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0answers
18 views

Weak Law of Large Numbers, biased expectation?

I want to show that: $$\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$ is a consistent estimator of $\sigma^2$. I was using the Weak Law of Large Numbers in the sense that: $$E(X_i-\bar{X })...
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0answers
63 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
3
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4answers
117 views

Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges

Suppose $a_0$ is an arbitrary positive real number. Define the sequence $\{a_n\}$ by $$a_{n+1}=\frac{a_n+2}{a_n+1}$$ for all $n\geq0$. I have to prove that $\{a_n\}$ converges. My attempt: If $a=\...
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1answer
36 views

Showing convergence of conditional probability

Let $\big(\Omega,\mathcal{F},\mathbb{P} \big)$ be a probability space, and $\big(E_n)_{n\in\mathbb{N}^*}$ such that $$ \mathbb{P}\big(E_n \mid Y \big) \underset{n\to +\infty}{\longrightarrow} 0\quad \...
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1answer
53 views

Sum of $resiprocals$ of the $Fibonacci$ $series$

Well I was having a doubt on the infinite sum of the reciprocals of the $Fibonacci$ $series$. That is: $S=1+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+....$ Assuming that the $series$ starts with $1$ ...
0
votes
1answer
20 views

Convergence in distribution: constant multiplication

If I have $$X_n/\sigma$$ converging in distribution to $\mathcal{N}(0,\sigma^2)$, does this mean I can just multiply through with $\sigma$ and obtain that $X_n$ converges to a standard normal? I ...
0
votes
1answer
52 views

Can Lebesgue Dominated Convergence always be used?

Suppose I want to find the derivative $$\frac{d}{dx}\int f(x,y) dy.$$ I want to know under what condition it would be equal to $$\int \frac{d}{dx}f(x,y) dy.$$ Of course, if I can find a suitable ...
0
votes
1answer
49 views

On computations related with $\lim_{x\to\infty} e^{-x}\sum_{\rho}\frac{(e^x)^\rho}{\rho}=0$

When I've reproduced the shape of the function $\sigma(x)$ of Apostol's section 4.10, a view of the page 98 is avaible as a Google Book (Apostol, Introduction to Analytic Number Theory, Springer 1976),...
0
votes
1answer
32 views

Convergence in probability of a sum of dependent random variables to 0 [closed]

Suppose we know that $A_n + B_n \xrightarrow{p} 0$ as $n \rightarrow \infty$, and we know that $A_n \xrightarrow{d} N(0,1)$. Can we say that $B_n \xrightarrow{d} N(0,1)$? Note that $A_n$ and $B_n$ are ...
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4answers
90 views

Knowing $1 = \lim_{k\rightarrow \infty} a_k + a_{k-1}$, does $a_k \rightarrow \frac{1}{2}$?

If I know that for all $k$ it holds that $a_k \geq a_{k-1} \geq 0$, and $1 = \lim_{k\rightarrow \infty} a_k + a_{k-1}$. Is this sufficient to conclude that $\lim_{k\rightarrow \infty} a_k = \frac{1}{2}...
3
votes
0answers
72 views

Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?

I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous. Statement:...
1
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1answer
33 views

Convergence of specific sequence in specific reduced group algebra equivalent to convergence of norms

Let $A = C_r^*(S_\infty)$ where $S_\infty$ - is permutation group of natural numbers fixing all but a finite number of element. Let $A_n = C_r^*(S_n)$ - subspaces of $A$ and $P_n : A \to A$ is ...