Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

4
votes
1answer
38 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
votes
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
vote
0answers
33 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.
0
votes
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
votes
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 + ...
0
votes
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
votes
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
votes
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
votes
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
167 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
39 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
24 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
votes
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
votes
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
vote
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
90 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
24 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
32 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
27 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
73 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
4answers
43 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
43 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
60 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 $n\rightarrow\infty$, ...
1
vote
3answers
51 views

Infinite series convergence test

Test the convergence of the following series: $${\sqrt{n+1}-1\over (n+2)^3 -1} +... \infty$$ (This is a problem I got on my test today, I constructed a similar series without the -1 part and showed ...
0
votes
1answer
34 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
0
votes
1answer
35 views

convergence/divergence $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$

I am stuck with this series $\sum_{n=2}^{\infty}(\frac{n}{-n+1})^n$. I used nth-root test, but the limit was $1$. Then I tried to think about it as $(-1)^n(\frac{n}{n-1})^n$ to use Leibniz, and I got ...
1
vote
0answers
34 views

Convergence of $\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$

I have to find for which $r\in\mathbb{R}$ a) series diverge b)converge absolutely c)converge not absolutely. $$\sum_{n=1}^{\infty}{r^n \cdot \sqrt n \cdot \arctan{\frac{1}{n+1}}}$$ I don't know ...
1
vote
2answers
41 views

$\sum_{n=1}^{\infty} \frac{(3n)! + 4^{(n+1)}}{(3n+1)!}$ converge or diverge?

does the serie $\sum_{n=1}^{\infty} \frac{(3n)! + 4^{(n+1)}}{(3n+1)!}$ converge or diverge? I tried this : $\sum_{n=1}^{\infty} \frac{(3n)! + 4^{(n+1)}}{(3n+1)!}= \sum_{n=1}^{\infty} ...
1
vote
1answer
22 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
1
vote
1answer
33 views

Generalized Poincaré Inequality

In my numeric script it states for a bounded domain $\Omega \subset\mathbb{R^n}$ and any map $0 < a_0 \leq a(x) \leq a_1 < \infty $ for $x\in \Omega$ it exists a $\gamma>0$ such that the ...
1
vote
1answer
55 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
0
votes
0answers
31 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...