Convergence of sequences and different modes of convergence.

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2
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5answers
191 views

Is integral convergent?

I have a problem with following integral: $\int_1^\infty \frac{\sqrt{x}}{1+x} \sin(2x)dx$ I was trying to prove convergence (or divergence) of this integral, however without any success. My best ...
5
votes
0answers
147 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
2
votes
4answers
223 views

Prove convergence of sequence defined recursively $a_{n+1} = \sqrt{6+a_n}$ [duplicate]

We have a sequence: $$a_1=\sqrt{6}$$ $$a_{n+1} = \sqrt{6+a_n}$$ The problem is to check convergence and then find the limit. We know that sequence converges when it is monotonic and bounded. With a ...
0
votes
1answer
51 views

Finding initial value to get a convergent sequence

Assume that a sequence of real number $\begin{Bmatrix} {S}_{n}\end{Bmatrix}$ satisfes: ${S}_{1}=b;\quad {S}_{n+1}={S}^{2}_{n}+\left(1-2a \right){S}_{n}+{a}^{2}(n\in \mathbb{N});\quad a,b\in ...
2
votes
1answer
52 views

Relations between convergence in nets and topologies.

I want to prove that given a net $S$ and topologies $T$ and $T'$, then $T\subset T'\iff$ when $S$ is convergent in $T'$ is also convergent in $T$. I'm proceeding this way: First I'd like to show that ...
3
votes
1answer
89 views

Will Fourier Series converge even if you only use Prime Integer frequencies?

So there is a Fourier Series for a function $f(x)$ with period $P$: $$ f(x) = \frac{A_0}{2} + \sum_{n=1}^{N} A_n \cdot \cos \left(\frac{n 2 \pi x}{P} + \phi_n \right) $$ Let $\frac{2 \pi x}{P} = t$ ...
1
vote
1answer
69 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
3
votes
1answer
39 views

Prove the inequality based on an infinite series

Define $$f(x)=\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}.$$ It is easy to see that this series converges for $x\in(-1,1).$ Now we are asked to show that $(1-x)^2f(x)\geq x,$ for $x\in[0,1).$ I tried ...
0
votes
0answers
36 views

mean square convergence vs almost sure convergence

I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge ...
2
votes
0answers
73 views

The value of a limit [duplicate]

Finding the value of the limit Let $ x_{n} $ =$ \sqrt{1+\sqrt{2+\sqrt{3+....+\sqrt{n}}}}$ then $\lim_{n\to\infty}x_{n}=$?
0
votes
1answer
40 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
2
votes
2answers
83 views

Sum of infinite series $\sum_{n=0}^{\infty}=\frac{n^2}{4n^2-1}t^n$

I have this problem, finding infinite sum of this series: $$\sum_{n=0}^{\infty}\frac{n^2}{4n^2-1}t^n$$ It should be done using derivatives and integrals, like for example: ...
0
votes
1answer
47 views

Suppose $\{x_n\}_{n=1}^{\infty}$ is a bounded divergent sequence. Let $S=$ range$\{x_n\}_{n=1}^{\infty}$. Can $S$ be an infinite set?

So $S$ is the set of all the terms of $\{x_n\}_{n=1}^{\infty}$. I feel like $S$ must be a finite set. I know that $\{x_n\}_{n=1}^{\infty}$ cannot be monotone, since it's bounded and divergent. So if ...
4
votes
1answer
56 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
3
votes
1answer
135 views

Dirichlet's function expressed as $\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)$ [duplicate]

How can we see that Dirichlet's function $$D(x):=\lim_{m\to\infty} \lim_{n\to\infty} \cos^{2n}(m!\pi x)= \begin{cases} 1 & x\in\mathbb Q\\ 0 & x\notin\mathbb Q\\ ...
2
votes
2answers
64 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
0
votes
0answers
33 views

A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
1
vote
0answers
100 views

Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
0
votes
4answers
74 views

Decide convergence of this series

How to prove the series $$\sum_{n=1}^\infty \frac {e^n\cdot n!}{n^n}$$ diverges? I tried D'Alambert and result is 1 and I'm stuck with Raabe.
0
votes
2answers
30 views

Let $u_n \to u$ in $L^1(\Omega)$. Does $u_n^p \to u^p$ in $L^1(\Omega)$ if we know $u_n^p \in L^1(\Omega)$?

Suppose $u_n \to u$ in $L^1(\Omega)$ where $\Omega$ is a bounded domain. Suppose that $u_n^p \in L^1(\Omega)$ (actually $L^\infty(\Omega)$ for each $n$). Fix $p \in [1,\infty)$. So $u_n(x) \to u(x)$ ...
0
votes
1answer
46 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
2
votes
1answer
58 views

Answer check on two series

I want to determine if these two are absolutely convergent, conditionally convergent or simply divergent. 1) $$\sum_{n=2}^\infty \left(\frac xn - \frac x{n-1}\right)$$ $$= \frac x2 - \frac x1 + ...
0
votes
2answers
62 views

How we compare the two following integral without calculation?

compare the two following integral without calculation : 1)$\displaystyle{\int_0^1x{e^{x^2}}dx}$ 2)$\displaystyle{\int_0^1 \sqrt{x}{e^{x}}dx}$ I would be interest for any comments or any replies
4
votes
2answers
118 views

L. Kronecker's theorem for sequences and series: $\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$

Assume $\sum a_i$ is a convergent series and $b_1,b_2,\dots$ is a divergent monotonically increasing sequence. How can we see that $$\lim_{n\to\infty}\frac{1}{b_n}\sum_{k=1}^na_kb_k=0$$ Attempt: We ...
0
votes
1answer
57 views

Prove or disprove the convergence of…

I need help with the following problem, please help. For positive real x. Let $${ B }_{ n }(x)\quad =\quad { 1 }^{ x }+{ 2 }^{ x }+{ 3 }^{ x }+...+{ n }^{ x }$$ Prove or disprove the convergence ...
0
votes
1answer
78 views

Cesaro summability together with $\lim nu_n\to 0$ implies convergence

Assume the series $\sum u_n$ is Cesaro summable and $\lim_{n\to\infty} nu_n\to 0$. We want to see that the series is (Cauchy) convergent. Attempt: Let $s_n=\sum_{i=1}^n u_n$ denote the $n$-th partial ...
0
votes
0answers
35 views

Convergence of two power series

I just wanted to know, whether my results are correct. I should find the radius of convergency in both cases: $\sum_{n=1}^\infty \frac{z^{2n}}{n^23^{n}}$ with a quotient criterion ...
2
votes
1answer
39 views

Sum of infinite series $\sum_{n=0}^{\infty}(-1)^n\frac{n-1}{n!}t^n$

I have this problem, finding infinite sum of this series $$\sum_{n=0}^{\infty}(-1)^n\frac{n-1}{n!}t^n$$ It should be done using derivatives and integrals, like for example: ...
2
votes
0answers
168 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
4
votes
1answer
55 views

Convergence of operator

I would like to know how to solve the following problem (since I didn't manage to solve it on today's exam): Let $A_h:L^1(a,b)\to L^1(a,b)$ be defined: $$A_h f(x)=\frac{1}{h}\int_x^{x+h} g(t) dt,$$ ...
2
votes
1answer
44 views

Determine the convergence of the infinite series [closed]

Determine the following infinite series is convergent or divergent. $$\sum_{n=1}^{\infty}\left[\left(1+\frac{1}{n}\right)^n-e\right]$$
0
votes
0answers
36 views

radius of convergence of hypergeometric functions

Hypergeometric function of scalar arguments is defined as \begin{eqnarray} _aF_b\left(p_1,...,p_a;q_1,...,q_b;z\right) &=&\sum_{i=0}^{\infty} ...
2
votes
1answer
28 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
0
votes
1answer
51 views

convergence radius of taylor series of a complex function in different directions, the same?

Given the taylor expansion of a complex functionf(z) around $z_0$, is the convergence radius of this series the same in different directions, say in real axis ...
2
votes
1answer
71 views

Are the measurable subsets closed under the symmetric distance metric?

Define the following pseudo-metric on the set of measurable subsets of $R$: $$D(A,B)=\operatorname{Length}((A\setminus B) \cup (B\setminus A)),$$ i.e., the distance between $A$ and $B$ is ...
11
votes
1answer
178 views

The integral on $[0,1]\times[0,1]$

Here I have a problem. $p$ and $q$ are positive numbers. the integral $$\int_0^1\int_0^1 \frac{1}{x^p+y^q}\;dx\;dy< \infty \Longleftrightarrow \frac{1}{p}+\frac{1}{q}>1$$ Here is my try, ...
2
votes
1answer
66 views

Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?

If $\displaystyle\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = ...
0
votes
2answers
31 views

Sums Convergence tests

$ \sum_{k=1}^\infty k(\frac 14)^k $ i've tried to do the D'Alembert's criterion and i got $ \frac 14 $ but according to wolfram alpha the answer is 4\9 thanks
5
votes
1answer
179 views

An interesting series to test convergence

I have another series in mind, today it is $$\sum^\infty_{n=10}\sin\left(\frac{1}{n^3}+\frac{\cos(n)}{n^2}\right)$$ I have tried to investigate the argument: it is basically $\frac{1+n\cos(n)}{n^3}$ ...
2
votes
2answers
73 views

Does this series with alternating elements converge?

I have to investigate convergence of series $$\sum_{k=10}^{+\infty}\frac{(-1)^k}{k+(-1)^k}$$ It certainly does not converge absolutely, because it is basically a harmonic series with every two ...
1
vote
1answer
179 views

Counterexample: Convergence in finite dimensional distributions does not imply weak convergence

I'm working at the following exercise: Give an example of a sequence of stochastic processes $(\mathbb{X}^n)_{n\geq 1}$ such that the finite dimensional distributions converge to the finite ...
0
votes
1answer
106 views

Convergence of $\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$

$\sum_{n=2}^{\infty}\frac{(-1)^n+\log(1+n^p)}{\sqrt{n-\sin n}}$ A) For which $p\in \mathbb{R}$ is the series convergent? B) For which $p\in \mathbb{R}$ is the series divergent, and what is ...
2
votes
4answers
98 views

How to prove this integral converge?

$$\int_{1}^{\infty }\frac{\ln x}{1+x^2}\,{\rm d}x$$ So far i tried to use the comparison test with $\int_{1}^{\infty }\frac{\sqrt{x}}{1+x^2}$ but i noticed that it's not always true. any ideas?
1
vote
1answer
31 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
2
votes
1answer
53 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
1
vote
1answer
28 views

Question about convergence

I have that $v=v^+-v^-$, $v^+,v^-$ are the positive and the négative part of $v$ and i have this: i dont understand why if $v_n\rightarrow v_0$ in $L^p(\Omega)$ then $v_n^+\rightarrow v_0^+$ in ...
0
votes
2answers
80 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...
0
votes
3answers
52 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
0
votes
1answer
51 views

Series convergence test of $\sum_{n=1}^\infty\frac{(1)}{n10^(n-1)}$

Given the following series I have to test the convergence. $\sum_{n=1}^\infty\frac{(1)}{n10^(n-1)}$. Then applying d'Alembert method I get: $\lim_{n\to\infty}\frac 1{(n+1)10^((n+1)-1)}\frac ...
0
votes
3answers
93 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...