Convergence of sequences and different modes of convergence.

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1answer
35 views

Why $N= \max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [closed]

Let $a_n=\dfrac{\sin(n)}{n^2 - 2}$. I am studying the proof of $\lim_{n\to\infty}a_n=0$. By definition: For every $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n ...
-4
votes
0answers
27 views

How can I prove or disprove that every series converges to philosophy? [on hold]

Let $x_0$ be any wikipedia page. Let $x_n$ be the page that the first non-etymological link on $x_{n-1}$ leads to. How can I prove that $x_n$ eventually converges to philosophy? Technically it seems ...
2
votes
0answers
10 views

Convergence of Sum of Random Variables “Independent in Limit”

Consider a sequence of random variables $X_n\sim U[-n,n]$, a random variable $Y\sim N(0,1)$, and a random variable $Z\sim U[0,1]$, all independently distributed. In addition, consider a bounded, ...
1
vote
0answers
29 views

Limit comparison test how to choose $b_n$?

$$\sum_{n=1}^\infty \frac{2n-1}{4n^2+1}\tag{1}$$ i would like to find out if this series convergent or not so i use Limit comparison test and choose $a_n$ and $b_n=\frac{1}{n}$ why do i need to ...
1
vote
4answers
28 views

Find the sequence of partial sums for the series $a_n = (-1)^n$ Does this series converge?

Find the sequence of partial sums for the series $$ \sum_{n=0}^\infty (-1)^n = 1 -1 + 1 -1 + 1 - \cdots$$ Does this series converge ? My answer is that the sequence $= 0.5 + 0.5(-1)^n$. This makes a ...
-1
votes
3answers
67 views

Determine whether the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ diverges or converges? - I want to check if my reasoning is correct

I got that the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ converges by doing the following: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ = $\sum_{n=0}^\infty ...
2
votes
1answer
13 views

Convergence in probability of product random variables

If $Y_n$s converge to constant $c$ in probability & $(X_n)$ is a sequense of random variables, is it true that $X_nY_n- cX_n$ converge to $0$ in probability? How can I prove this? Thanks in ...
1
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3answers
52 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $(x_n)$ tending to $0$ such that $f(x_n)=0$ for all $n$, then $f(x)=0$ for all $x$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
1
vote
2answers
25 views

Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
2
votes
7answers
85 views

Tell if a sum is convergent $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$

$$\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$$ I tried to solve this by saying that $$\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}$$ I then made two sums like this: $$\sum\limits_{n=1}^\infty ...
1
vote
1answer
43 views

A question about integrable function.

My question: Let $(\Omega, \mathscr{U}, \mu)$ be a measure space, and let $X$ be an integrable function and let $A, \ \ \{A_n\} \in \mathscr{U}; n\in \Bbb N$. How to prove that $$\int_{A_n} X d\mu ...
0
votes
1answer
58 views

A question about improper integral

Would you please give me a hint on how to solve this problem: Suppose $f(x)$ continuous in $[0,\infty)$ and for each $a,b>0$ and $c>b$, we have \begin{equation*} ab \left|\int_0^1 f\left(ax+c ...
0
votes
1answer
13 views

Question about a bounded sequence in $W^{1,1}$ admitting a weakly convergent subsequence

Let $B$ be the unit ball in $\mathbb{R}^{12}$ and $f_n\in L^7(B)$ such that $\lVert f_n \rVert_{W^{1,4}(B)}$ is bounded. Is it true that there exists a subsequence weakly convergent in $W^{1,1}(B)$? ...
2
votes
1answer
29 views

Existence of a global limit in $L^1([-N,N])$ for each $N\in \mathbb{N}$

Let $(f_n)_n$ be sequence of functions $f_n\in L^1_{loc}(\mathbb{R})$ such that for each $N\in \mathbb{N}$, $(f_n)_n$ is a Cauchy sequence in $L^1([-N,N])$. Then for each $N$, $(f_n)_n$ converges to a ...
0
votes
0answers
13 views

Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used. (crosspost from crossvalidated).

I have asked this question on crossvalidated here and I am still unsure on the answer. I attempt a cross-post (most of the times this proves very useful). I copy the question below: I am trying to ...
-2
votes
1answer
21 views
3
votes
1answer
51 views

Show $\sum e^{-nx + \cos(nx)}$ is defined on $(a, \infty) $ for any $a>0 \dots$

I want to prove that $\sum e^{-nx + \cos(nx)}$ is defined and continuous on the given interval of $(a, \infty)$ where $a >0$. Then, how exactly do I show it is defined? It just seems trivial, ...
5
votes
1answer
30 views

Proof check, showing pointwise convergence

My problem is this: For $x \in [0,\frac{\pi}{2}]$, $f_n(x) = \frac{nx}{1 + n\sin(x)}$ Find the pointwise limit of $(f_n)$ for all $x \in [0, \frac{\pi}{2}]$ I am not sure if the way I constructed ...
1
vote
1answer
28 views

Determining if $\int f_n\to 0$ implies that $f_n\to 0$ in measure and $f_n(x)\to 0$ a.e.

If $f_n$ is a sequence of measurable functions on $(X,\mu)$ into $[0,1]$ and $\int f_n\to 0$, I am trying to prove (or disprove the following): (i) $f_n$ converges to $0$ in measure. (ii) For almost ...
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0answers
27 views

If the sequence of random variables $X_n$ converge in Probability to $X$ and $Y$, then $X = Y$ a.s. [on hold]

If the sequence of random variables $X_n$ converge in Probability to $X$ and $Y$, then $X = Y$ a.s. Idea: I want that $P(|X-Y|> \epsilon) = 0$, for every $\epsilon >0$. $P(|X-Y|> ...
2
votes
1answer
66 views
+100

Legendre Differential Equation, $y_1,y_2$ linearly independent solutions

$$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$ At the interval $(-1,1)$ the above differential equation can be written equivalently $$y''+p(x)y'+q(x)y=0, ...
5
votes
5answers
64 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
1
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0answers
40 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
1
vote
1answer
39 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
1
vote
2answers
38 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
0
votes
1answer
24 views

Riemann's Lemma in proof

Consider the following expression: $$s_n(x) - s = \frac{1}{2\pi} \int_{-\pi}^{\pi} h(t) \exp(i\frac{1}{2}t)\exp(int) \ dt - \frac{1}{2\pi} \int_{- \pi}^\pi h(t) \exp(-i\frac{1}{2}t)\exp(-int) \ dt $$ ...
2
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0answers
25 views

Convergence of a series of integrals

Definition of problem I would like to find the large $N$ behaviour of a summation series and specifically the conditions under which it converges. I have found one condition, but I think it might be ...
0
votes
3answers
24 views

Show convergence of 1/cosh series

Sorry if my english is not correct. Feel free to edit and ask questions. I need to test the following series on convergence: a) $$ \sum_{n=0}^{\infty}\frac { sinh(n) }{ e^n } $$ and b) $$ ...
0
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0answers
31 views

Determine the radius of convergence of the power series

Determine the radius of convergence of the power series $\sum \limits _{n=4} ^\infty \frac {2n+4} {4^{n+5}} (x-8)^{4n+1}$. I tried the ratio test to find where $\frac {a_n} {a_{n+1}} < 1$ but I ...
0
votes
0answers
22 views

Pointwise convergence of Bernstein polynomials for piecewise continuous functions

I know that $B_nf \to f$ uniformly if $f:[0,1] \to \mathbb R$ is continuous. But can anybody explain to me, why $B_nf \to f$ pointwise in every point where $f$ is continuous if $f:[0,1]\to \mathbb ...
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votes
2answers
84 views

How to prove that $f_n(x)=\frac{nx}{1+n\sin(x)}$ does not converge uniformly on $[0, \pi/2]$? [duplicate]

If $f_n$ is a sequence of functions over $[0, \pi/2]$ given by $$f_n(x) = \frac {nx} {1+n\sin(x)},$$ then how would I go about proving that $f_n$ does not converge uniformly to a function $f$ on ...
0
votes
1answer
29 views

Interval of convergence of power series?

If the power series is: $$ \sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n+1}} $$ and I've found the interval to be $$ -1 < x < 1 $$ then would the answer $$ (-1, 1) $$ work? some other options ...
0
votes
1answer
37 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
2
votes
3answers
30 views

(Simple question) Radius of covergence of power series?

For the power series: $$ \sum\limits_{n=1}^{\infty}\frac{(x-1)^n}{2^n} $$ Would radius of convergence be $$ x = 1 $$ ?
2
votes
2answers
57 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
2
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1answer
41 views

What is the center of power series?

The power series is: $$ \sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1} $$ Any help appreciated!
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2answers
1k views

Why is the ratio test for $L=1$ inconclusive?

One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or ...
1
vote
1answer
17 views

Multidimensional convergence in probability

If I have a vector $X^n=(X^n_1,...,X^n_m)$ is it true that $ \mathbb{P}(X^n\geq\epsilon)\rightarrow 0$ if $ \mathbb{P}(X^n_i\geq\epsilon_i)\rightarrow 0\ \forall i =1,...,m$ As $n\rightarrow \infty$?
0
votes
1answer
37 views

Example of a convergent series for which integral test fails?

Is there example of a convergent series for which integral test fails or can not be applied? Just wondering if integral test is the silver bullet of convergence tests, or are there any series that any ...
0
votes
2answers
25 views

Sums of converging limits

How can I prove the property that if the sequences, $(x)\rightarrow x' $ and $(y)\rightarrow y'$ then $(x) + (y)\rightarrow x'+y'$
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votes
2answers
60 views

Is $(-1)^{n!}$ convergent? [on hold]

I don't think I can use the alternating series test because of the factorial sign, but I don't know how else to solve this. can you please give any hints ?
0
votes
0answers
12 views

Deriving Dual Averaging from (Sub)gradient Descent

Here the presenter tries to derive a simple Dual Averaging from (sub)gradient descent. I have a little problems understanding the steps. (Sub)gradient descent: Loop through: $$ x_{k+1} = x_k - t_k ...
2
votes
0answers
26 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
2
votes
1answer
17 views

Cauchy sequence of partial sums of orthogonal vectors in a general Hilbert Space.

Let $(x_n)$ be a sequence of orthogonal vectors in a Hilbert space $(V, \langle,\rangle)$. For $n = 1, 2, 3, ... $ put, $$s_n = \sum_{j=1}^{n} x_j.$$ (a) Calculate $\|s_n\|$ in terms of ...
3
votes
0answers
51 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
0
votes
1answer
13 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
-2
votes
0answers
16 views

Convercenge in probability implies convergence in Lp [on hold]

Show that if $X_n$ is that $|X_n|< C$, with $C\in \mathbb{R}$, $\forall n \in \mathbb{N}$, then $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
2
votes
1answer
38 views

Convergence of complex power series $z^{n!}$ at boundary

I'm revising for an exam at the moment and I'm struggling with part of a question. I'm asked to find the radius of convergence of the series $\sum_{n=0}^{\infty }z^{n!}$ and then find where it ...
1
vote
0answers
11 views

Understanding the difference between convergence in distribution and convergence almost surely

I know that the sum of $\sum_{i=0}^nZ_i$ where $Z\sim N(0,1)$ has a distribution of a Chi squared distribution with $n$ degrees of freedom which in my understand means that $Z^2$ converges in ...
2
votes
1answer
20 views

Pointwise convergence - $\frac{nx}{1+n \sin(x)}$ , $x \in [0, \frac{\pi}{2}]$

Is anyone able to check if this is correct: for $$f_n(x) = \frac{nx}{1+n \sin(x)} , x \in [0, \frac{\pi}{2}]$$ Does this converge pointwise to $$ \frac{x}{\sin(x)}$$ I am unsure due to the fact ...