Convergence of sequences and different modes of convergence.

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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 ...
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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 \...
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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
60 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. $$\sum_{n=3}^\infty \frac{\tan\left(\frac{\pi}{n}\right)}{n}$$
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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
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1answer
129 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
76 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 })...
0
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1answer
71 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 ...
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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 $1$ ...
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1answer
21 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 ...
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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 ...
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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),...
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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}...
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0answers
73 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:...
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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 ...
3
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1answer
50 views

Universal change of variable $t = \tan(\frac{\theta}{2})$

In many textbooks, the authors explain, without rigorous justification, we could always solve a trigonometric integral in using the change of variable $t = \tan(\frac{\theta}{2})$. Is there anyone ...
2
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0answers
72 views

Taylor series around $x = 0$ of $f(x) = \int_0^x\frac{dy}{1+y^4}$

Find the Taylor series around $x = 0$ of $f(x) = \int_0^x\frac{dy}{1+y^4}$ and its radius of convergence. It seems like one ought to take the Taylor series of the integrand, and then integrate the ...
3
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2answers
47 views

if $f_n(x)$ converges uniformly to a function $f(x)$ does $f_n'(x)$ converge uniformly to $f'(x)$?

Let [a,b] denote a finite interval and consider a sequence $\{f_n(x)\}_{n=0}^\infty$ in $C^1([a,b])$. if $f_n(x)$ converges uniformly to a function $f(x)$ on $[a,b]$, does $\{f_n'(x)\}$ converge ...
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1answer
66 views

Are there general methods that can be applied when using the Borel-Cantelli Lemma, to get a statement about a sequence of random variables?

I hope the title in itself is clear, if not allow me to give an example. In Class my Professor did the following: Given a sequence $(X_n)_{n \in \mathbb{N}}$ of non-negative i.i.d. RV $X_n \sim X$...
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1answer
23 views

Convergence of specific sequence in reduced group algebra

I have $B = C_r^* (S_\infty)$ - reduced group algebra of permutation group of naturam numbers fixing all but a finite number of element. $B$ have countable family of subalgebras: $B_n = C_r^* (S_n)$ ...
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0answers
14 views

Use operator norm to rigorously prove exp(ln(I + A)) = I + A

Show that $\exp(\ln(I + A)) = I + A$ when the operator norm of $A$ is less than 1. A similar question has been posted, Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?, but this does not offer a real ...
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1answer
45 views

How to prove $\sum_{r=1}^\infty \frac{(r-1)π ^2}{\sqrt{1+(rπ )^4}}$ diverges

Using comparison test with the series $\sum_{r=1}^\infty \frac{1}{r}$. The above series diverges. But how to show the comparison exactly? $$1+(rπ)^4 < {((rπ)^2+1)}^2 ; \frac 1 {\sqrt{1+(rπ)^4}} &...
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2answers
54 views

Finding convergence zone/range for $\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$

$$\sum_{i=1}^\infty \frac{x^{n^2}}{n(n+1)}$$ I used the ratio test and I end up with: $$|x|*\frac{n}{n+2}$$ What steps do I need to take to continue? Looking for hints or steps, not full solution/
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1answer
21 views

Definition of convergent sequence in topological space

The book I'm reading says: If $\{x_n\}$ is a sequence of points in $(X,T)$ and $x \in (X,T)$, the sequence is said to converge to $x$ iff for each open set $U \in T$, there exists an $n_0 \in \Bbb N$ ...
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1answer
36 views

Sequence Cauchy w.r.t $L^2$-norm but has no convergent subsequence

I have a problem with following exercise: Let $C([0,1])$ be the space of continuous real valued functions on the closed unit inverval $[0,1]$. And let the $L^2$-norm given by: \begin{equation} \Vert f\...
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1answer
18 views

Vasicek equation [closed]

Vasicek interest rate stochastic differential equation is $$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$ where $\alpha , \beta$ are positive constants. I need to use Ito-Doeblin formula to compute $...
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1answer
142 views

Weyl's asympotic law for eigenvalue : $\lim_{\lambda \to \infty} \frac{M(\lambda)}{\lambda}=\sum_p \frac{A(D_p)}{4 \pi}$

In the book Strauss W.A. Partial differential equations - An introduction (Wiley, $2008$, $1$nd Ed.) page $310 - 311$, it is probably a silly question, but is there anyone could give me a hint how to ...
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0answers
26 views

Quadratic covariation

I am trying to solve small task from stocastic calculus. It can be shown that for stochastic processes X and Y , the quadratic covariation satisfies the polarisation formula - $[X, Y](T) = 1/2 ([X + ...
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1answer
19 views

Convergence of the inverse in Sobolev spaces

Assume we have a sequence $f_k$ which converges to $f$ in the Sobolev space $H^p(D)$, where $D\subset\mathbb{R}^N$ ($N\geq 2$) is relatively compact and $p\geq 1$ is an integer. We also assume that $$...
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1answer
36 views

Using of Ito formula with martingales

We have exam test - $\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$. I know that $\beta$ can be chosen so that $N$ ...
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1answer
34 views

Proving that $\frac{nx}{2+n+x}$ converges uniformly on $0 \le x \le 1$

Proving that $f_n(x)=\frac{nx}{2+n+x}$ converges uniformly on $0 \le x \le 1$ Now I know I have to use the infinity metric, but I can't understand the solution given for this question. The next ...
2
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1answer
14 views

Graph the describes the peaks of a Poisson Distribution

Probably not a very smart question - barely know any of the more interesting parts of probability theory - but I noticed that the peaks of Poisson curves form what looks like a kind of logarithmic ...
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1answer
26 views

Ito integral for simple stochastic process

I need for $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I know that Ito integral for simple stochastic process satisfies $E[I^2 (t)]=E\int_0^t\Delta^2(u)...
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0answers
45 views

Can mathematics help a poor researcher?: ODE, convergence, solution without an exact solution

General Firstly, I would like to say that it is my first time using Math StackExchange. So, if the written format is not the commonly accepted format, I do apologise and I would be very happy if you ...
0
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1answer
36 views

Why can't an argument for the Riemann Zeta function be 1? What happens if we take Re(s)=1? [duplicate]

If $s=1$, then the series equals to $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...\to \infty$ This certainly does seem to be a convergent series. Why doesn't it have a limit?
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2answers
61 views

Testing convergence of series $\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$ [duplicate]

Considering $$\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$$ depending on $k$, which can be real. I have absolutely no clue how to proceed. Tried to taylor it, but with no result.
0
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1answer
36 views

Testing convergence of a series $\sum_{n=3}^\infty\ln\left(\frac{\cosh(\pi/n)}{\cos(\pi/n)}\right)$

Given series $$\sum_{n=3}^\infty\ln\left(\frac{\cosh(\pi/n)}{\cos(\pi/n)}\right)$$ I should test convergence. I know, that I should use a comparison criterion. I tried expressing $$2\cosh(x)=e^x+e^{-x}...
2
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1answer
28 views

radius of convergence of ${3^{k^2}}{x^{k^2}}$

find the interval of convergence $\Sigma _{k=0} ^{\infty} 3^{k^{2}} x^{k^{2}}$ The radius of convergence of this series is 1/3 by the book. but the answer and what i think is so different what i ...
2
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1answer
37 views

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$?

Does $\lVert S_Nf-f\rVert_{\infty}\to 0$ imply $\lVert S_Nf-f\rVert_{L^2}\to 0$ I have a proof for the first case, under the assumption that $f$ is $C^1$ and real valued (also $1$ periodic) $\lVert ...