Convergence of sequences and different modes of convergence.

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Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
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0answers
13 views

if a sequence converges in measure in $L^p$, then converging for weak topology.

Given a finite measure space $(A,\Sigma,\mu)$, for $p \in (1,\infty)$, if {$f_n$} is a bounded sequence in $L^p(A)$ converging in measure to $f \in L^p (A)$, then {$f_n$} converges to $f$ for the ...
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4answers
49 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
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1answer
20 views

Root Test for Convergence or Divergence (ln problem)

I'm stuck at this problem. How would I approach this (I have to use the root test for this one)? The ln and e and my power is throwing me off. This is how far I have gotten.
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1answer
15 views

Interval of convergence for a series

I am currently trying to determine the interval of convergence, but I keep getting 0 for all my questions. I have attached one of the questions that I am unable to solve completely and I would really ...
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1answer
8 views

Asymptotic convergence of the total length of a graph

I encoded the following algorithm: suppose we're in (0,1)x(0,1) and I randomly create a "village" one at a time. At each step, I link a newly randomly created village to the closest village already ...
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2answers
51 views

radius of convergence of $\sum_{n=1}^\infty n!^2x^{n^2}$ [on hold]

Determine the radius of convergence of the following power series: $\sum_{n=1}^\infty n!^2x^{n^2}$
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1answer
23 views

Almost uniform convergence implies a.e. pointwise convergence proof

I've just read a proof of the statement "On a finite measurable space, $(f_n)_{n \geq 1}$ and $f$ measurable and finite a.e. functions, if $(f_n)_{n \geq 1}$ converges almost uniformly to $f$, then it ...
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2answers
21 views

Use ratio test to test for convergence or divergence

I have online hw and it tells me if my answer is correct or not. It said that my answer for this problem is incorrect: Can someone tell me what I did wrong? Also I might be asking alot of these ...
1
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1answer
41 views

Showing convergence and divergence

Say I have: $(x_n)$ a sequence of real numbers such that $\sum x_n$ which converges conditionally and implies $\sum x_{2n}$ diverges. I want to show that $x_{2n}$ does not in general converge. So I ...
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2answers
43 views

Understanding complex numbers

I need to show that $$\left | \sum_{k=1}^n e^{ik}\right | $$ is bounded Now I am given that $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ But have little idea of how to proceed further and ...
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1answer
29 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
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1answer
46 views

Question about convergence in $L^2$

Assume we have a sequence of functions $\{f_n\}_{n\geq 0}\subset L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$, i.e. $$\lim_{n\to \infty} \int_0^1 |f_n(x)-f(x)|^2dx =0.$$ Is it then true ...
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1answer
46 views

Does $\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$ converge?

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$$ Because this is an alternating series, I decided to use the alternating series test. This ...
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4answers
41 views

$d_n=(1+(2/n))^n$ converge or diverge and find the limit?

I know the answer is $e^2$ and I'd like to use L'Hopital's rule because this is an indeterminate form. Can someone explain how to get there? $$d_n=\left(1+\frac{2}{n}\right)^n$$
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2answers
30 views

Convergence of infinite series from 2 to infinity 1/(x((lnx)^2))

On a recent exam I was asked to test the following series for convergence From $2$ to $\infty$ $\frac{1}{x(lnx)^{2}}$ I blanked on the integral but set up a comparison test, saying that ...
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1answer
30 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...
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2answers
40 views

How to prove that a sequence converges

I am having some trouble understanding how I can show that a given series converges. I found a general explanation here that states: To prove that a sequence converges, it is sometimes easier to ...
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0answers
22 views

Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
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0answers
14 views

How would I prove the following convergence? [duplicate]

I found the following code in an algorithms book and can't really seem to prove why it returns a square root, I understand it intuitively, but can't prove it. ...
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0answers
26 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...
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1answer
45 views

Riemann-Stieltjes Integrability and Convergent Series

Let $\alpha_{n=1}^{\infty}$ be a sequence of monotonically increasing functions on $[a.b]$ such that the series $\sum_{n=1}^{\infty}\alpha_{n}(a)$ and $\sum_{n=1}^{\infty}\alpha_{n}(b)$ converge. ...
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3answers
239 views

Absolute convergence to zero imply convergence to zero?

Given that $\frac{1}{N}\sum_{i=1}^N {|a_i|}$ converges to zero as $N\rightarrow \infty$, does it imply that $\frac{1}{N}\sum_{i=1}^N {a_i}\rightarrow 0$? I know absolute convergence imply ...
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2answers
15 views

Conditionally convergent sequences and implications

If I have $\sum b_n$ is conditionally convergent, how can I show that $\sum b_{4n}$ doesn't in general converge? Assume $(b_n)$ is an arbitrary sequence of the Reals All I need is a counter example ...
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1answer
25 views

proof related to convergence of a integral

i have the following condition $$0\le f(x)\le g(x)$$ and $$\int_{a}^{b}g(x)dx$$ is convergent for any $a$ and $b$ (which means $a$ or $b$ can tend to infinity) then prove that $$\int_{a}^{b}f(x)dx$$ ...
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2answers
29 views

How to prove if a series converge

I need to prove if this series converge and if yes, prove the partial sum. I have no idea where to start Can somebody tell me the steps i need to follow?
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5answers
67 views

Determine whether the series $\sum_n \frac{1}{n^n}$ converges or not? [closed]

I believe I need to use the ratio test to prove it is convergent.
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2answers
24 views

Convergence of a sequence of matrices

Let $A$ be a $n×m$ matrix with real entries, and let $B = AA^ t $and let $\alpha$ be the supremum of $x ^t Bx$ where supremum is taken over all vectors $x ∈ \mathbb R ^n$ with norm less than or equal ...
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1answer
19 views

The convergence set of a sequence of functions can be expressed in terms of upper and lower envelopes

let $f_n:\mathbb R\to[0,\infty)$ be a sequence of functions. Its lower envelope sequences are defined as $\underline{f_n}(x)=\inf\{f_k(x):k\geq n\}$. And its upper envelope is defined similarly except ...
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2answers
53 views

If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if ...
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3answers
60 views

Convergence of sequence $x_{n+1}=\sqrt{2x_n+3}$

Let $f:\Bbb R \rightarrow \Bbb R$ be the function $f(x)=\sqrt{2x+3}$. (a) Show that for all $x\in[1,3], f(x)\geq x$. You may use the intermediate value theorem if you like. Let $x_0=1$ ...
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1answer
41 views

Convergence to infinity of a sum of independent random variables

I am doing an exercise which says: Suppose $(X_n)$ is a sequence of independent random variables (not necessarily identically distributed) with finite variances. Write $S_n:= \sum_{i=1}^n X_j$ for ...
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1answer
29 views

Convergence related to Dominated convergence

Let X : (Ω, F, P) → (R, B) satisfy E[|X|] < ∞. If $A_n ∈ F$ is a sequence of sets with $lim_{n\rightarrow \infty} P(A_n) = 0$, then prove that $lim_{n\rightarrow \infty} E[|X|1_{A_n} ] = 0$. Here ...
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3answers
86 views

Determine whether the series $\sum_n\frac{n-1}{2^{n+1}}$ is convergent. [closed]

I know I need to write it as a telescoping sum. Also, compute the limit if it exists.
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1answer
39 views

Prove $\cos(n_k x)$ does not converge pointwise on $[0,2\pi]$

There are strictly increasing sequence $n_1<n_2<\dots<n_k<\dots$ which are positive integers. I want to prove on the domain $[0,2\pi]$ where $\cos(n_kx)$ converges does not coincide with ...
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1answer
29 views

Show a series of functions is discontinuous at a point

I have a series of functions which converges to an integrable function. I need to show that this function is discontinuous at every point . For starters (because of the way it's defined) I'm just ...
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1answer
23 views

Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...
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1answer
50 views

How can I show $| \sum_{k=1}^n e^{ik}|$ is bounded?

I know that we can write $ \sum_{k=1}^n e^{ik} = \frac{e^{i(n+1)} -1}{e^i - 1}$ But I am unsure how to proceed with showing there's some $M \in \mathbb{R}$ where $\forall n \in \mathbb{N} \space ...
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1answer
16 views

Iterated Limits Along an Ultrafilter

Setting: Let $\mathfrak{U}$ be an ultrafilter on an index set $I$. Let $G$ be a compact group with identity $e$, and let $\mathbb{T}$ denote the unit circle in the complex plane. For each $i\in ...
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1answer
17 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
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21 views

decimal representing a positive real number is a Convergent series [closed]

Prove that every decimal representing a positive real number can be expressed as a convergent series.
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1answer
22 views

A relation between convergence in measure and pointwise convergence

Let $\{f_n\}$ be a sequence of measurable functions on $R$ with Lesbegue measure and $f$ be a measurable function. I have to show that $\{f_n\}$ converges to $f$ in measure if and only if any ...
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1answer
51 views

Exponential of a matrix always converges

I am trying to show that the exponential of a matrix converges for any given square matrix of size $n\times n$: $M\mapsto e^M$ e.g. $\displaystyle e^M = \sum_{n=0}^\infty \frac{M^n}{n!}$ Can I argue ...
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0answers
23 views

find all the values of x such that the given series would converge in interval notation [closed]

find all values of $x$ such that the given series would converge $$\sum_{n=1}^{\infty}\frac{(8x-2)^n}{n^2}$$ Give your answer in interval notation
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2answers
117 views

How can I prove that this recursive sequence converges?

Let $a_0=1$, $a_1=1$ and $a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$. How can I prove that this sequence is convergent? I know that if it's convergent, it converges to $\sqrt{2}$ and I can calculate the ...
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1answer
25 views

Convergence of random variables in $L^1$

So $g$ is a continuous real-valued function and are given that the sequence of random variables $Y_n$ converges to $Y$ in $L^1$, $E[|g(Y_n)|]<\infty$ and $E[|g(Y)|]<\infty$. Show that $g(Y_n)$ ...
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1answer
23 views

Convergence of the Riemann zeta function in $\mathbb Q_p$

Does the Riemann zeta function without p-Euler factor i.e. $\prod\limits_{\text{prime }q \not= p}\frac{1}{1-q^{-1}}$ converges in $\mathbb Q_p$?
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1answer
84 views

Almost sure convergence of the series of independent random variables

Let $\{X_n:n\ge1\}$ be i.i.d. random variables with $\operatorname EX_1=0$ and $\operatorname E|X_1|^p<\infty$, where $1<p<2$. Let $\{b_n:n\ge1\}$ be a real sequence. Does the series $$ ...
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1answer
46 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...
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0answers
12 views

Multivariate Delta Method

If I have a $\sqrt{N}$ asymptotic normal estimator (call it $\boldsymbol{\theta}$, possibly a vector). Say I want to find the asymptotic distribution of $g(\boldsymbol{\theta})$ and suppose ...