Convergence of sequences and different modes of convergence.

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What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
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1answer
25 views

Define sequence and convergence

Define function f: $\mathbb{R}_+\rightarrow\mathbb{R} $ by: $ f(x)=\sqrt{\frac{x^2}{3}+\frac{18}{x}}$ 1) Show that $f'$ has one minimum/maximum, define $f'$s monotony conditions and sketch $f$. I ...
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3answers
202 views

Proving the sum of the reciprocals squared converges [duplicate]

I'm investigating the Basel Problem, and the sum to consider is: $\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}$ How can I show this converges? Using graphs/computer software is also fine, but ...
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1answer
35 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
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1answer
17 views

Convergence of the sequence of maxima of a function sequence

Suppose we have a compact set $K \subset \mathbb{R}$ and a sequence of continuous functions $f_n: K \rightarrow \mathbb{R}$. Let $f$ be the uniform (and hence continuous) limit of $(f_n)_n$. Assume ...
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1answer
29 views

Show that $f_n(x) = \log nx$ does not converge pointwisely

I am learning convergence of sequence of functions in $\mathbb{R}$ and I would like to show the following: Define $f_n : (0,1) \rightarrow \mathbb{R}, f_n(x)=\log nx$. Then $\{f_n(x)\}$does not ...
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1answer
16 views

Convergence of $a_n$ given $\limsup\frac{\log{a_n}}{\log{n}}$ [on hold]

Let $$p=-\limsup\frac{\log{a_n}}{\log{n}}$$ Show that $p$ makes for a good convergence test. That is, $\sum a_n$ converges if $p>1$.
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5answers
40 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
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0answers
28 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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3answers
57 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
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1answer
21 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
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1answer
58 views

Does the statement “$f = 0$ almost everywhere” depend on the measure that is defined?

I know the convention is to use the Lebesgue measure but is there ever a situation where we would interpret "$f(x) = 0$ almost everywhere" by using a different measure? For example, let $f(x) = 1$. ...
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0answers
29 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
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1answer
16 views

Convergence of the maxima of Cauchy random variables

Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of independent and identically distributed random variables such that $X_1$ has density function $f_{X_1}(x) = \frac{1}{\pi(1+x^2)}$, $x \in ...
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1answer
26 views

Product of a Convergent Series and Bounded Sequence

Let $a_n$ be a bounded sequence and $\sum_{n=1}^\infty b_n$ be a convergent series. Then $\sum_{n=1}^\infty b_na_n$ is convergent. I have found a counterexample to prove it false; If we let ...
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1answer
35 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
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2answers
29 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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0answers
24 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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1answer
25 views

Yet another interchange of limit and sum (revised)

A revised version of a similar question I asked before. $S_n$ are finite sets containing rational numbers in [0,1] for which $S_n\subset S_{n+1}$ and $\lim_n S_n = \mathbb{Q}\cap [0,1]$. ...
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1answer
37 views

Yet another interchange of limit and integral

$f_n$ are uniformly-bounded functions that converge pointwise on a dense subset of the compact set $D$ to a continuous function $f$. Is it true that $$\lim_{n\to\infty} \int_D f_n(x)\, \mbox{d}x = ...
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3answers
129 views

Show sequence is convergent and the limit

Given the sequence $$\left\{a_n \right\}_{n=1}^\infty $$ which is defined by $$a_1=1 \\ a_{n+1}=\sqrt{1+2a_n} \ \ \ \text{for} \ n\geq 1 $$ I have to show that the sequence is convergent and find ...
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2answers
50 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
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1answer
31 views

Prove that $\lim_{n\to\infty} f_n=0$ on $[0,\pi],$ where $f_n(x)=\cos^n(x)$

Prove that $\lim _{n\to\infty} f_n=0$ on $[0,\pi],$ where $f_n(x)=\cos^n(x)$ It's not difficult to see that when $x\in(0,\pi)$, we can take some large $N$ such that $\forall \epsilon>0$, $n\ge ...
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1answer
25 views

Convergence in Distribution

Suppose $X_n$ are $\mathbb{R}$-valued random variables. Then how would I be able to show that $X_n\rightarrow^D X$ where $X$ is also a $\mathbb{R}$-valued random variable iff $F_n(x)\rightarrow F(x)$ ...
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1answer
18 views

Alternating Series Test for Divergence

Alternating Series Test if the alternating series $$\sum^\infty_{n=1}(-1)^{n-1}b_n=b_1-b_2+b_3-b_4+b_5+...\;\;\;\;\;b_n\gt0$$ satisfies $$(\text{i)}\;\;b_{n+1}\leq b_n\;\;\;\;\;\text{for all ...
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2answers
70 views

Another way to find $\sum_{n=1}^{\infty}{\dfrac1{k^n}}$

Find sum $$S=\sum_{n=1}^{\infty}{\dfrac1{k^n}}$$ where $k\in\mathbb{Z^+}\setminus\{1\}$. In my book they used limits to find this sum. They wrote $$S=-1+\sum_{n=0}^{\infty}{\dfrac1{k^n}}$$ and then ...
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1answer
31 views

Showing that $X_n$ ~ $N(0, a_n)$ converge to $0$ when $a_n \to 0$ sufficiently fast

If $X_n$ have distribution $N(0, a_n)$ with $\sum_{n=1}^\infty a_n^b < \infty$ for some $b > 0$, then $X_n$ converge almost surely to $0$. I was able to show (for a previous part of the ...
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1answer
37 views

Series convergence and limit

I am trying to solve an exercise that asks me this: Prove using the $ε–N$ method that the sequence $a(n) = \frac{n^2 + n - 1}{n^2 + n}$ converges and state the limit. My attempt is the following: ...
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1answer
25 views

Finding a function greater or less than factorial function

suppose we are given the sequence: $a_n = (-1)^n\frac{1}{n!}$ using squeeze theorem find the limit: $$\lim_{n\to \infty} (-1)^n\frac{1}{n!}$$ using the squeeze theorem. For factorials, $a_n$ how ...
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0answers
41 views

The convergence of the multiplication of two convergent series? [on hold]

If we know that \begin{equation} {\sum\limits_{n=1}^{\infty} }a_n \end{equation} and \begin{equation} {\sum\limits_{n=1}^{\infty} }b_n \end{equation} are convergent What about their ...
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Determine if $\displaystyle \int_3^{\infty}\frac{x+1}{\sqrt{x^4-x}}\,dx $ converges

Determine whether the following integral is convergent or divergent without evaluating it. (Whichever answer is correct, you must show why it is true.) $$ ...
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4answers
104 views

Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?

$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$ The first one is an alternating series, so it would just be: $$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow ...
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1answer
29 views

convergence of an alternating series without $a_n$

I have an infinite series that goes like: 3-$\frac{69}{5}$+$\frac{834}{25}$-$\frac{7734}{125}$+$\frac{62109}{625}$-$\frac{455859}{3125}$+$\ldots$ I can generate more terms of this series if needed. ...
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3answers
46 views

What is the difference between convergence of a sequence and convergence of a series?

I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" I understand the following: ...
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2answers
109 views

Covergence test of $\sum_{n\geq 1}{\frac{|\sin n|}{n}}$

I need to prove that $$\sum_{n\geq 1}{\frac{|\sin n|}{n}}$$ is convergent. How should I do it?
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4answers
51 views

Determining whether $\sum_{k=1}^\infty \frac{x^k}k$ converges [on hold]

$$\sum_{k=1}^\infty \frac{x^k}k$$ Does this series converge, if yes, then for what values of $x$?
2
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2answers
54 views

Evaluate $\lim_{n\to\infty}\prod_{k=1}^{n}\frac{2k}{2k-1}$

How can I calculate the following limit: $$ \lim_{n\to \infty} \frac{2\cdot 4 \cdots (2n)}{1\cdot 3 \cdot 5 \cdots (2n-1)} $$ without using the root test or the ratio test for convergence? I have ...
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3answers
38 views

Show the space X is incomplete

A problem comes from the "Optimization by vector space methods". Luenberger p.34 Let $X$ be 1. the space of continuous functions on [0,1] 2. its norm is defined by $||x|| = \int^1_0|x(t)| dt$ So, 1. ...
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4answers
221 views

Does $\sum_{x=1}^\infty\sin(x)$ converge?

I received a task to find out whether the following series converges: $$\sum_{x=1}^\infty\sin(x)$$ On first look it seems simple, but as I keep thinking about it, there's not a single lemma or ...
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1answer
29 views

Convergence of a simple Integral

Let $f$ be continuous on the interval $[0, 1]$. How can I show that $$\lim_{n\to\infty}\int_0^1 f(x)\sin nx\,dx = 0 ?$$
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0answers
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Convergence Pointwise or Uniformly? [closed]

I just want to check which of these converge pointwise or uniformly? fn(x) = sin(x)/n, x is a real number. fn(x) = 1/(1+nx), x is in [0,1]. fn(x) = x/(1 + n*x^2), x is a real number. Thanks
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0answers
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Proving convergence of a martingale in $L^2$ [closed]

I'm stuck with the following problem: Let $X$ a positive martingale bounded in $L^2$. Show that $\lim_{n\to \infty} X_n = X$ a.s. and in $L^2$.
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2answers
23 views

Convergence of a sequence of functions depending on a sequence of points

Let the sequence $(x_k)_{k\in\mathbb{N}}$ converge to $x$, where $x_k\in D\subset \mathbb{R}$. Let $(f_k)_{k\in\mathbb{N}}$ be a sequence of functions from $D$ to $\mathbb{R}$ such that $ ...
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0answers
20 views

conditional expectation convergence in L1 [closed]

Let $\mathcal{F}_n \uparrow \mathcal{F}_{\infty}$ and $Y_n \rightarrow Y$ in $L^1$. I'm stuck on how to show that $E(Y_n | \mathcal{F}_n ) \rightarrow E(Y | \mathcal{F}_{\infty})$ in $L^1$?
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1answer
47 views

Show that the sequence of products $\prod_{k=1}^n (1+1/k^3)$ converges

$$ a_{n} = 1 + \frac{1}{n^3} $$ Show that the sequence is converges $$ \lim_{n \rightarrow \infty} \left(1 + \frac{1}{1^3}\right)\left(1 + \frac{1}{2^3}\right)\left(1 + \frac{1}{3^3}\right) \ldots ...
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1answer
94 views

Determine $p$, $q$ and $r$ so that the order of the fixed point iteration for computing $a^{1/3}$ becomes as high as possible

So I'm given the following equation for computing $a^{1/3}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this equation converges to $a^{1/3}$ as ...
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1answer
30 views

Let $a_n \rightarrow a$. Show that $\liminf(a_n-b_n)=a-\limsup(b_n)$.

Assignment: Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences of real numbers with $a_n \rightarrow a \in \mathbb{R}$. Show that: $$\liminf(a_n-b_n)=a-\limsup(b_n)$$ ...
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0answers
10 views

Finding the range of x for which a particular sum converges.

I am doing some practise questions which are going to help towards my exam but I am stuck on this one question. Can anyone solve the question and tell me all the steps to the final solution.
0
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1answer
24 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
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1answer
38 views

How can I prove that a sequence such that every converging subsequence coverges to the same limit, converges?

I want to claim that if $(x_n)_{n\in N}$ is a sequence, and there is $a$ such that if $(x_{n_k})$ converges, so $\lim x_{n_k} = a$ (it means that all converging subsequences have the same limit), then ...