Convergence of sequences and different modes of convergence.

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1answer
19 views

Convergence in probability of a random variable

I need to prove that $(X_n^2 -X)^2\to 0$ in probability $\Rightarrow X_n^2\to X$ in probability. I tried solving it with the triangle inequality, but it didn't get me anywhere. Is there another ...
2
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5answers
171 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
104 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.. ...
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0answers
30 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
2
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4answers
59 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
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1answer
26 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
36 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
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1answer
49 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
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0answers
36 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
36 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 ...
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0answers
19 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
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0answers
68 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
28 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
71 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
31 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
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1answer
39 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, & ...
1
vote
1answer
62 views

Dirichlet's function

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
58 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
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0answers
30 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
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0answers
35 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
62 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.
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2answers
24 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
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1answer
40 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
42 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 + ...
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0answers
22 views

Solve for values that satisfies convergence of a product of two functions is infinite.

I need to find the values of $\gamma>0$ that satisfies, that $$\lim_{x \to \infty} ( \exp(\theta x)x^{-\log(x)^\gamma}) = \infty \; \forall \; \theta > 0 $$. My gut tells me that this will ...
0
votes
2answers
58 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
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2answers
89 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
41 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
43 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
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0answers
32 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
30 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
96 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
52 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
37 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
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0answers
21 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
21 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 ...
1
vote
1answer
40 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 ...
1
vote
1answer
32 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
168 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, ...
1
vote
1answer
40 views

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

If $\{f_n(x)\}$ is sequence of continuous function on $\mathbb{R}$ converging uniformly to $f(x)$,then does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) ...
0
votes
2answers
25 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
137 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 $\frac{1+n\cos(n)}{n^3}$ ...
0
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0answers
14 views

Searching for theorems that prove almost sure convergence from convergence in probability

As we can see that almost sure convergence implies convergence in probability, and the converse is not necessarily true. But now I would like to prove a particular sequence of random variable ...
2
votes
2answers
55 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 ...
0
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0answers
27 views

Proof that feature normalization cause faster convergence of gradient descent

How to prove that if I do feature normalization (scaling of the $x_1,\ldots,x_n$ to be all in range $[0,1]$) to a convex function $f(x_1,\ldots,x_n)$ that returns real scalar, then gradient descent ...
1
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0answers
13 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
73 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
91 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?
0
votes
0answers
23 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
0
votes
1answer
25 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 ...