Convergence of sequences and different modes of convergence.

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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
19 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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56 views

Convergent series and real numbers [closed]

Prove that every decimal representing a positive real number can be expressed as a convergent series. Any ideas?
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1answer
23 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 ...
2
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1answer
61 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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2answers
123 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 $g(Y_n)$ ...
2
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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 $\mathbb Q_p$?
2
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1answer
89 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
50 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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13 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 ...
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0answers
34 views

Root Test when testing convergence

When using the root test, does the absolute value of the limit have to be less than 1 or does just the limit have to be less than positive 1? ...
0
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1answer
28 views

Convergence test via integral

I've got to the problem of testing convergence using the integrals on $$ \sum _{n=1} ^{\infty} \arcsin \left( \frac{1}{\sqrt{x}} \right) $$ Our theory says: Consider an integer $N$ and a ...
2
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2answers
32 views

Pointwise convergence and L1 convergence in bounded mass case

I have a question regarding convergence modes and their relationships, my problem is actually an application to probability, How to prove that : ${f_n}$ and $g$ are probability density functions such ...
3
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1answer
40 views

Rate of Convergence of complicated sequence with interactions

I have been working on a problem where the sequence turns out to be so complex that i am unable to find its convergence rate with necessary and sufficient conditions on the parameters.After working ...
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1answer
19 views

Estimating convergence order

For a homework problem, we are told to perform the first four iterations of the Newton-Raphson method for a function This part I have no problem with. It then asks us to "check the error decrease ...
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1answer
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solution of a convergent equation? [on hold]

I have a equation that don't know how to solve: 1. $f(1)>0.437 $ 2. $f(x_i+1)>f(x_i) $ 3. $\sum(f(x_i))=43.7$ 4. $x_i=i; i=1,2,3...,100 $ what is $f(x)$?
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97 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
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1answer
49 views

Deduce alternate series test from Abel’s theorem

Show that the alternate series test can be deduced from Abel’s theorem. I know that Abel's theorem is Abel's Theorem Let $(a_n)$ and $(b_n)$ be two sequences of real numbers such that • $(a_n)$ ...
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46 views

Determine whether the following sequence ${a_n}$ is convergent

Determine whether the following sequence $(a_n)$ is convergent, stating its limit if it exists. $$a_n=\frac{6n^3+2n+(-4)^n}{4^n-1}$$
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Weird behavior of Non-Lebesgue measurable subset of the Smith–Volterra–Cantor set and pointwise convergence of a sequence of simple functions

I came across this (seemingly?) weird behavior of a sequence of simple functions: Let $E$ be the Smith–Volterra–Cantor set and $m: \operatorname{Leb}(\Bbb{R}) \to [0, \infty]$ the Lebesgue-measure. ...
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1answer
42 views

Existence of a sequence that converges to infimum of a function

X is a compact subset of $\Re^{n}$. A upper and lower bounded function f is defined that f:X $\rightarrow$ $\Re$. Does a sequence of {$x_n$} always exist so that f($x_n$)$\rightarrow$ $\inf$[f(x)]. ...
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96 views

Show the following series converges to $\frac{1}{e}$.

Let $\tau \in \mathbb{N} $ whereas $\tau$ fulfils the following condition: $$ \sum_{i= \tau}^{n-1} \frac{1}{i} \lt 1\le \sum_{i=\tau-1}^{n-1} \frac{1}{i}$$ Prove that $$\lim_{n \to \infty} ...
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1answer
20 views

Help with series convergence and divergence concept

When you are trying to figure out whether a series converges or diverges why can you test for convergence at any term in the series as opposed to having to start at the beginning of the series. Is it ...
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0answers
17 views

How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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0answers
20 views

When can weak law of large numbers be turned into a strong law?

I am trying to do the following exercise from Chung. I would like a hint if possible, rather than a full solution. Let $(X_n)$ be a sequence of random variables (not necessarily independent or ...
3
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1answer
38 views

When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I ...
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36 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
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1answer
38 views

Series Summation,Convergence

I am stuck on the 4 th one.I have done the rest.I have found out the value of a_n.But not getting how to proceed for the 4 th one.
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Definition of Strong Convergence in $L^p$

Is strong convergence in $L^p$, ie $f_i \overset{strongly}\longrightarrow$, just $||f_i-f||_{L^p} \rightarrow 0$. If so why dont we just call it convergence?
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Unordered convergent sums

In Elementary Real Analysis of Thomson and Bruckner, I'm stuck in exercise 3.3.4 page 88. Could you please help me giving me hints? Let the infinite index set $I$. Show that if $\sum\limits_{i\in ...
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34 views

Two questions about Almost sure convergence and Uniform integrability

Let $X_n$ and $Y_n$ be two sequences of random variables such that $X_n\stackrel{n}{\rightarrow}C$ almost sure and $Y_n\stackrel{n}{\rightarrow}C$ almost sure, $C\in \mathbb{R}$. Suppose that $X_n$ ...
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55 views

$\sum a_n$ converges conditionally

If we assume that $\sum a_n$ converges conditionally then How can we comment that $\sum a_{2n} $ does not converges, While it does when $\sum a_n$ converges absolutely ?
2
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1answer
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If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e then $f_n \rightarrow f$ uniformly on each $E_j$

If $\mu$ is $\sigma$ finite and $f_n \rightarrow f$ a.e, there exists $E_1,E_2, \ldots \subset X$ such that $\mu((\bigcup_{1}^{\infty}E_j)^{c})=0$ and $f_n \rightarrow f$ uniformly on each $E_j$ My ...
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1answer
65 views

Prove sequence converges to zero

There are two sequences $(a_n)$ and $(b_n)$ and I know that their multiplication $(a_n b_n)$ converges to 0. Let there be a constant $c>0$ that for almost every $n$, $b_n\geq c$. I need to prove ...
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2answers
101 views

$\sum a_{2n} $ converges [closed]

Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n} $ converges. I know this part, How it can be done but i am having problem in proving the later part of the question T o show that this ...
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2answers
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Lack of understanding of the proof of $\sum\limits_{i\in\mathbb{Z}}2^{-|i|}=3$.

In Elementary Real Analysis of Thomson and Bruckner p85 the proof of $\sum\limits_{i\in\mathbb{Z}}2^{-|i|}=3$ is given: I didn't understand why:$$\sum_{j\in J}2^{-|j|}<2(2^{-N})$$ Could you please ...
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1answer
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How to show that Newton's method has the linear convergence rate with 1-1/m?

How to show that Newton's method has the linear convergence rate with 1-1/m ? (For a zero of multiplicity m>=2)
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Strong law of large numbers with changing expected value

Question: Suppose that $X_1^n,...,X_n^n \sim^{iid} X^n$ and $X^n \rightarrow X$ in distribution/weakly. Is it true that $\frac{1}{n} \sum_{i=1}^{n} X_i^n \rightarrow \mathbb{E}[X]$ almost surely? ...
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2answers
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Finding the $p,\ r,\ q$ for which the series converges

I'm dealing with the series: $$\sum_{n=3}^{\infty} \frac{1}{n^p(\ln n)^q(\ln(\ln n))^r},$$ looking for the set of all $p,q,r$ such that the series converges. Is there a way to determine this without ...
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Does $\sum_{n=1}^\infty\sin(n)\sin\left(\frac{\pi}{2n}\right)$ converge?

I must determine whether if the following series converges, converges absolutely, or diverges: $$\sum_{n=1}^\infty\sin(n)\sin\left(\frac{\pi}{2n}\right)$$ By the comparison test, I have already found ...
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1answer
95 views

When does $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ absolutely converge?

Let $p>0$. I must find the values of $p$ for which the following series converges: $$\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$$ I have already successfully proven the following estimate by induction: ...
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2answers
22 views

Is this a valid step in a convergence proof?

I'm asked to say what the following limit is, and then prove it using the definition of convergence. $\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$. Is it valid to say that the sequence ...
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1answer
38 views

Finding convergence parameter

Find the value of the parameter $p$ for which the following series converge. Series of: $\frac{1}{(ln(k))^p}$ from $2$ to $\infty$.
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92 views

Simple convergence proof

I'm asked to prove, using the definition of convergence, that limits approach a certain value. For example, $$\lim_{n\rightarrow\infty}\dfrac{n^2+4}{n^2}.$$ I can see that it converges to $1$, but ...
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47 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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Convergence of Sequences Proof

Let $(x_n)$ and $(y_n)$ be convergent sequences. Use the definition of convergence (no limit theorems!) to prove that the sequence $(3x_n2y_n)$ converges. I'm having trouble doing this using the ...
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1answer
48 views

Determining a general sequence is bounded and has a particular limit.

I'm having a hard time with the following problem. I seem to be going around in circles in trying to prove either. Some tips in finding a proof for both would be appreciated. For $n\ge 1$, let ...
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0answers
24 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...