Convergence of sequences and different modes of convergence.

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$0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

this is related to that one the limits of $a_n $and $b_n$ Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
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45 views

the limits of $a_n $and $b_n$

this is related to that one $a_n$ is bounded and decreasing Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
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58 views

$a_n$ is bounded and decreasing

my second question from [An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$ Let $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and ...
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43 views

$b_{n}$ is increasing

I think there is misunderstanding in my last post because its contain three questions so i will post question by question step by step An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi ...
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147 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
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1answer
34 views

If $(f_n)_n$ and $(g_n)_n$ converge stochastically to $f$ and $g$, then $(f_n+g_n)_n$ converges stochastically to $f+g$

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space $(E,d)$ be a separable metric space $f,g,f_n,g_n:(\Omega,\mathcal{A})\to(E,\mathcal{B}(E))$ measurable $(a_n)_{n\in\mathbb{N}}\subseteq E$ We ...
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Sequence and series problem

How do I show that the sequence $(x_n)$ defined by $$x_ {n+1} = \left(1-\frac{1}{n}\right) ^2 x_n + \frac{1}{n}, \forall \,n \in \Bbb{N}-\left\{0\right\}$$ converges? and to what limit?
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36 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
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2answers
44 views

If $X, X_1, X_2, \ldots $ are positive and $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$

Let $X, X_1, X_2, \ldots $ be positive random variables. Prove that if $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$ My attempt: I tried to truncate $E(|X_n-X|)$ ...
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3answers
64 views

find sum of ln(1- $\frac{1}{n^2})$. prove it converges [duplicate]

![enter image description here][1] $$ \sum = \ln(1 - \frac{1}{4} ) + \ln(1 - \frac{1}{9} ) + \ln ( 1 - \frac{1}{16}) = \ln(\frac{3}{4}) + \ln (\frac{8}{9}) + \ln ( \frac{15}{16}) = -.287 + -.117 + ...
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When does this integral converge? $\;\;\int_0^\infty {\frac{e^{-ax}}{x^2+1}}\,\mathrm{d}x$

$$\int_0^\infty \frac{e^{-ax}}{x^2+1}\,\mathrm{d}x$$ - $a$ real, for which a does this converge? (The final answer is $a\ge 0$) I've tried doing this by parts and it seems to work at first, but then ...
5
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1answer
79 views

Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent?

Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$. Is the following series always ...
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0answers
20 views

Is this a viable generalization of Newton series?

I wonder if the following formula a viable generalization of Newton's series. $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty ...
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1answer
38 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
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1answer
35 views

Question about convergence proof, why has he chosen the parameter this way

In this proof he says that $n > 2k$, but would it work if $n \ge k$, if not, why? If $p>0$ and $\alpha$ is real, then $\displaystyle\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. Proof: ...
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5answers
78 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
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1answer
85 views

Integral of the convolution of two functions: $\int_{-\infty}^{\infty} (f*g)(x)dx$

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
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1answer
62 views

What is the radius of convergence of the power series $\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$?

What is the radius of convergence of the power series? $$\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$$ Progress Used the ratio test, but got $0$ from it.
2
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1answer
19 views

Convergence in probability given that covariance matrix goes to $0$

Suppose I have a sequence of random vectors $\{X_n\}$ each of dimension $2\times 1$. Suppose also that I know $$ ...
0
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1answer
69 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...
2
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3answers
83 views

Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
4
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1answer
125 views

Show that series converge or diverge

If $\displaystyle \sum_{n=1}^{\infty} a_n$ converge and has positive terms then decide if following series converge or diverge : a) $\displaystyle \sum_{n=1}^{\infty} a_n \cdot \sin{a_n}$ I think it ...
3
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2answers
61 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
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1answer
92 views

What's the limit of this sum $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ [closed]

Let $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ and s be a complex variable . $s=\sigma +it $ where $\sigma ,t \in\mathbb{R} $ , Note :I edit the ...
2
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2answers
71 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
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1answer
43 views

Applying the dominated convergence theorem to $\lim_{n\to\infty} x^n$, for $x \in [0,1]$.

$\lim\limits_{n\to\infty} x^n$, for $x \in [0,1]$. I'm using the dominated convergence theorem on a few problems and keep running into this issue. What's the limit of the above function? ...
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1answer
71 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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2answers
36 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
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1answer
34 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
2
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2answers
50 views

If $\{x_n\}$ is a sequence in $\mathbb{N}$ and $x_n \rightarrow x$, prove there exists $N$ such that $x_n = x$ for $n \geq N$

Since $x_n \rightarrow x$, we know that for all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - x| < \epsilon$. We want to show that for some $\epsilon ...
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66 views

Convergence of series of $1/n^x$ - pointwise and uniformly,

Consider the series $$\zeta(x) = \sum_{n\ge 1}\frac {1}{n^x}.$$ For which $x \in[0,\infty)$ does it converge pointwise? On which intervals of $[0,\infty)$ does it converge uniformly? My work: I ...
2
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1answer
42 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
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0answers
36 views

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for ...
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1answer
59 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
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1answer
54 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...
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55 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
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1answer
20 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
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1answer
37 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
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1answer
47 views

Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
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1answer
37 views

$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
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1answer
19 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
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1answer
21 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
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1answer
56 views

What is the solution to poissons equation with a point source?

I am currently trying to solve the following PDE using various numerical methods - direct and iterative. $$ \nabla ^2 u = \rho $$ where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The ...
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1answer
43 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
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0answers
31 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
0
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1answer
37 views

What is the difference between the limit of a sequence and a limit point of a set?

I always thought they were the same thing. The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point ...
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1answer
53 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
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3answers
33 views

Can someone help me understand the proof that every cauchy sequence is bounded?

This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I ...
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2answers
80 views

Don't understand proof that $x_n \rightarrow A$ $\iff$ every subsequence of $\{x_n\}$ converges to $A$

So we are given that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, we have that $|x_n - A| < \epsilon$ and we want to show that for all $\epsilon' > 0$, there ...
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1answer
88 views

A step in proving that a real Cauchy sequence is convergent.

I'm trying to prove that a real Cauchy sequence is convergent, but I need some help for a step. We have the following statements: $\{ s_i\}$ is a real Cauchy sequence, i.e. ...