Convergence of sequences and different modes of convergence.

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59 views

dominated convergence theorem application

Prove or disprove this statement: if $f_n : \mathbb R \to \mathbb R$ are integrable functions with $f_n \to 0$ pointwise and $|f_n(x)| \le \frac{1}{|x| + 1}$ for all $n$, $x$, then $\lim\limits_{n ...
0
votes
2answers
45 views

Almost sure convergence of $\max(X_1, X_2,\ldots,X_n)$.

$X_1, X_2,\ldots, X_n$ are independent with uniform distribution on $[0,a]$. Prove that $\max(X_1, X_2,\ldots,X_n) \rightarrow a$ almost surely. Ar first I look for the probability distribution i.e. ...
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0answers
35 views

Limit of integrals of simple functions over a finite measure

We are given a sequence of simple functions $f_n:\mathbb{R}^2\rightarrow \mathbb{R}$ which converge pointwise to a continuous limiting function $f$. We also have a bunch of positive, finite measures ...
1
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1answer
28 views

How do I find the interval of convergence?

Suppose I have: $$\sum \cfrac{(-1)^n}{\sqrt{n}}x^n$$ If I use the ratio test, I get $$\cfrac{1}{\sqrt{1+\frac{1}{n}}}|x|$$ Why can it be said the radius of convergence here is $1$? Disregard the ...
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0answers
16 views

Limits and Convergence of sequences in the form of $(k, k^2, 1/k)$

I'm dealing with proving the convergence and limits of sequences that are defined by multiple points, such as $$ \left(k, k^2, \frac{1}{k}\right) $$ and I'm not sure how to go about doing it. I'm ...
6
votes
1answer
169 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$?

How do I evaluate this sum :$$\displaystyle\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$$ Note : I used many criterions of convergence to show if it converges but i didn't up. Thank you for any ...
7
votes
4answers
550 views

Integral of odd function doesn't converge?

When I look up $$\int_{-1}^1 \dfrac{1}{x} dx$$ on Wolfram Alpha, it says it doesn't converge. While this is a sum of two diverging integrals, the two areas are clearly symmetric, and I'd assume the ...
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1answer
24 views

Convergence of Series for tangent (only convergence or divergence)

$$\sum_{n=17}^{\infty}\left(\tan\left(\frac{1}{n}\right)\right)^2 \ \ $$ My first guess is to write the series as integral. And use the substitution for u=1/n. That changes my upper and lower ...
1
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1answer
37 views

Prove completeness of a metric space

Let $\mathcal{K} = \{A \subset \mathbb{R}^N| A \neq \emptyset, A \text{ closed and bounded with respect to the euclidean metric} \}$ Let us define $A_\epsilon = \bigcup_{x \in A}U_\epsilon(x)$, where ...
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2answers
54 views

Find radius of convergence for the given sequence: $\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$

I've been trying to realize how to find the radius of convergence for this sequence: $$\sum_{1}^{\infty} \frac{(-1)^n}{n!}x^n$$ I know that it converges for any given $x$, but can someone explain me ...
5
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2answers
58 views

Show that the sequence $\langle b_n\rangle$ Converges to $1$

The following question was asked in my Masters entrance examination but unfortunately I was unable to answer this. Please tell me how to approach this problem correctly. Suppose $\langle ...
0
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0answers
16 views

correlated random vectors converge to an independent random vector

Suppose $(X_k,Y_k)_{k\geq 1}$ is a sequence of random vectors. $X_k,Y_k$ have support $[0,1]$ and a joint density function $ f_k(x,y)$ being positive for all $x,y\in [0,1]$ and continuous everywhere. ...
2
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3answers
69 views

Integral test for convergence of $\frac{1}{\ln x}$ [closed]

I want to know if $$\int_0^1 \frac{1}{\ln x}\, dx$$ converges or not.
5
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3answers
61 views

Determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$

I tried to use D'Alambert theorem to determine convergence of the series $\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}$ . $\lim_{n \to \infty} \frac{a_{n+1}}{a_{n}} = \lim_{n \to \infty} ...
0
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1answer
53 views

Determine convergence of the series $\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$

How to determine convergence of the series: $$\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$$ I spent most of the time using the Integral criteria (since the function $f(x)=\frac{1}{\ln(x)^{\ln(x)}}$ ...
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1answer
26 views

Relation between sum of indicators and average of probability for independent sequence

Let $A_1, A_2, \ldots$ be independent events, set $N_n := \sum_{i=1}^n I_{A_i}$ and $\overline p_n := n^{-1} \sum_{i=1}^n P(A_i)$, then $$ P\left( \lim_n \left( n^{-1} N_n - \overline p_n ...
2
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3answers
25 views

Conditionally convergent - limit of series'

Let $\sum_{n=1}^\infty a_n$ be conditionally convergent. Let $k_n:= \max(a_n,0),l_n:=-\min(a_n,0)$ for $n\in \mathbb{N}$ and show that $\sum_{n=1}^\infty k_n =\infty $ and ...
2
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0answers
34 views

Describing convergence with probability $1$ in “finite” terms, proof correct

I tried to solve the following exercise: Show that $Z_n \to Z$ with probability $1$ if and only if for every $\varepsilon$ there exists some $n$ such that $P(|Z_k - Z| < \varepsilon, n \le k ...
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0answers
14 views

Rearranging series' to converge to a certain point.

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
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0answers
14 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
0
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3answers
32 views

Trouble with the proof of convergence of a series.

$\sum_{n=1}^\infty \frac{n}{(-2)^n}$ I tried using D'Alembert's Ratio on it and this is how far I got: $\frac{(n+1)}{(-2)^{n+1}}\frac{(-2)^n}{n}=\frac{n+1}{(-2)\cdot ...
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0answers
26 views

Rearrangement of series' - not absolutely convergent [duplicate]

Let $\sum_{n=1}^\infty a_n$ a convergent but not absolutely convergent series. 1.1: For $n\in \mathbb{N}$ let $b_n:=max(a_n,0)$, $c_n=-min(a_n,0)$ Show that: $\sum_{n=1}^\infty ...
1
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1answer
28 views

Convergence in probability implies boundedness in $L^1$?

Suppose we have a sequence of positive random variables wgich converges to 0 in probability, i.e. $X_n=o_P(1)$. I want to prove that $E[X_n]$ is bounded. My idea: In particular $X_n$ is bounded in ...
3
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1answer
62 views

The Passare-Tsikh solution to the principal quintic

The Bring-Jerrard quintic, $$x^5+x+t=0$$ can be solved as, $$x = -\sum_{k=0}^\infty(-1)^k\frac{(5k)!}{k!(4k+1)!}\;t^{4k+1}\tag1$$ when, $$|t|<\frac{4}{5^{5/4}}\approx 0.53\dots$$ This paper ...
0
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4answers
73 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
-1
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0answers
54 views

Product of divergent and convergent sequence

I have the following question: Let $\{a_n\}$ be a sequence of positive real numbers that converges to $a.$ Find the value for $a$ so that the series ...
0
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1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
0
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1answer
28 views

Snowflake-sequences - Area - Circumference

Consider the following inductively defined snowflakes-sequences: $S_1$ is an equilateral triangle with edge length $l_0$, and $S_{n+1}$ emerges from $S_n$ by dividing each edge by 3 and the middle ...
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2answers
35 views

Series' - Convergence - Limit of sequences

Examine the following series' for convergence: a)$\sum_{n=1}^\infty \frac{n^3\cdot 3^n}{n!}$, b)$\sum_{n=1}^\infty\frac{n}{(-2n)^n}$, c)$\sum_{n=1}^\infty \frac{n!}{n^n}$, ...
1
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3answers
41 views

Convergence of the series $\sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right)$ [closed]

Does the series $$ \sum_{n\in\mathbb N}\left(\sin\frac{1}{n^n}\cdot 2^n\cdot n!\right) $$ converge?
2
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4answers
94 views

How would you prove this converges? $\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$

How would you check if this converges or not? $$\sum_{1}^{\infty} \frac{(-1)^n}{[n-(-1)^n]^{\frac{2}{3}}}$$ It looks like a telescopic sequence so I thought I'd first write the beginning values: ...
4
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4answers
148 views

How to prove that $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$

I have this sequence: $$\sum_{1}^{\infty} \frac{1}{n^3}$$ and I need to prove that: $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$ So basically I know that this sequence converges using the integral ...
0
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1answer
33 views

Law of large numbers for nonnegative random variables [closed]

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. ...
-1
votes
1answer
97 views

prove the convergence sequence [closed]

How to prove that the sequence is convergent? ${a_{n+2} = {\sqrt{a_{n+1}} + \sqrt{a_{n}}}}$ , where $a_{1}=1$ and $a_{2}$ is a positive number.
3
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1answer
25 views

Example of non-commutative infinite product of complex numbers.

I have read a proof of the following theorem in Rudin's Real and Complex Analysis: Suppose $\{u_n\}$ is a sequence of bounded complex functions on a set S, such that $\sum |u_n(s)|$ I converges ...
1
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2answers
57 views

How to solve this sequence?

I have this sequence: $\sum_{n=1}^{\infty} \frac{n^2+n-1}{\sqrt{n^\alpha+n+3}}$ For which values of $\alpha$ does this converge? I first tried to separate into cases where $\alpha \gt 0$ etc and ...
3
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1answer
56 views

Does $\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0$ imply $u\in L^2(\Omega)$?

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $u\in C^0(\Omega)$ and $(u_n)_{n\in\mathbb{N}}\subseteq C_0^0(\Omega)$ with $$\left\|u_n-u\right\|_{L^2(\Omega)}\stackrel{n\to\infty}{\to}0\tag{1}$$ Can ...
0
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2answers
49 views

product converging to one

I have a question concerning an infinite product. Suppose $x_n$ is a sequence of positive real numbers. My intuition says that $$\lim_n(1-\exp(-x_n))^n=1$$ for any sequence $x_n=n^\alpha$ with $\alpha ...
2
votes
1answer
36 views

Show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s.

Given a sequence $(X_n)_{n\geq 1}$, show that $|X_n|\leq Y$ a.s. implies $\sup_n |X_n|\leq Y$ a.s. Here is my attempt: $|X_n|\leq Y$ a.s. means that $P(|X_n|>Y)=0$, $\forall n\geq 1$ $P(\sup_n ...
2
votes
1answer
35 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
0
votes
2answers
33 views

How can I use the sequential criterion to prove/disprove the existence of a limit?

In my book (Abbott), it is written that the sequential criterion for the limit of a sequence is as much of a tool to prove as it is to disprove the existence of a limit. My question is: how? For ...
3
votes
1answer
61 views

Examples and counter-examples in Real analysis - check my answers please

I've been given some practice examples, without solutions in preparation for an upcoming exam, and was hoping I could get them double checked here. For each of the following, either give an example ...
0
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0answers
30 views

How to interpret this statement about two sequences?

Say, $(a_n)_n, (b_n)_n$ are two sequences of non-negative real numbers and both converge to zero. Moreover, we know that $$\forall \varepsilon,M > 0 \ \ \ \exists n_0 \in \mathbb N \ \ \ \forall n ...
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0answers
20 views

Verifying Integrals with convergence theorems

I would like to check with you guys, whether my solutions are correct so far. Problem: Verify following statements $\lim\limits_{n\rightarrow\infty} \int_0^\infty e^{-nx} \sin(e^x) dx = 0$ ...
2
votes
1answer
80 views

Verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0.$

How to verify $\lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0$ My idea is to use the dominant convergence theorem with $f_n(x):= e^{-nx} \sin(e^x)$ and ...
0
votes
1answer
21 views

Convergence radius

I know the Cauchy Hadamard equation to calculate the convergence radius of a power series $$\sum_{n=0}^{\infty} a_n x^n$$ Is there a way to generalize this for series of the form ...
0
votes
1answer
10 views

The majorant/minorant criterion

The majorant criterion says if a series in a Banach space has a convergent majorant, then it converges absolutely. My question is, what if a series in a Banach space has a convergent minorant, does it ...
0
votes
0answers
18 views

Analysing the convergence of improper integral with parameter

Can you, please, check if it's right what I did: Here's the exercise: Test the convergence of the following improper integral which is defined using parameter $p\in R$: ...
2
votes
1answer
59 views

$\sup_{t \in [0, \infty)} \left|[(H^{(n)} - H) \cdot X, Y]_t \right| \overset{P}{\rightarrow} 0$

1. Notation We start with establishing some (standard, I think) notation. Let $(\Omega, \mathcal{A}, P)$ be a given probability space. For any filtration $\mathcal{G} = (\mathcal{G}_t)_{t \in ...
1
vote
2answers
41 views

How do I prove that this sequence converges? $\sum_{0}^{\infty} \frac{n-3}{n+2}^{n^2-n}$

I've been having trouble checking whether this sequence converges or not: $$\sum_{n=0}^{\infty} \frac{n-3}{n+2}^\left({n^2-n}\right)$$ At a first glance I thought I should try the root test but that ...