Convergence of sequences and different modes of convergence.

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3
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5answers
63 views

$a_n = {{2^n}\over n}$ diverges? [closed]

How do I formally show that the sequence $a_n = {{2^n}\over n}$ diverges using a $\delta$-$\epsilon$ argument?
2
votes
1answer
26 views

Continuous mapping theorem to show $g(x_n)$ to $g(x)$ not converges.

Let $X_n$ be a random variable sequence, such that $P(X_n=1)=1/n$ and $P(X_n=1/n)=1-(1/n)$. Let g be a function, such that $ g(x)= 0$ if $ x\le0$, and 1 if $x>0$ Show that $g(X_n)$ not converges ...
2
votes
0answers
40 views

Limit of a function is the function of the limit (no continuity or convergence)?

Suppose I have a sequence of real-numbers $\{a_n\}_n$ and that $\lim_{n \rightarrow \infty}a_n=a$ where $a$ can be finite or infinite. Consider a function $f(\cdot)$ of $a_n$. Under which conditions ...
0
votes
0answers
12 views

which one is the correct way for finding radius of convergence?

which the correct one for finding radius of convergence ? My professor teach me that $r =\frac{a_n}{a_{(n+1)}} $ but in internet it says $r =\frac{a_{(n+1)}}{a_n} $ can someone give some ...
0
votes
2answers
38 views

Proof by induction for recursive sequence with no explicit formula.

The problem I am trying to solve is: "show that the sequence defined by $a_1=6$ and $a_{n+1}=\sqrt{6+a_n}$ for $n\ge 1$ is convergent, and find the limit." So I know that I need to use proof by ...
1
vote
2answers
18 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
2
votes
5answers
79 views

Limit of the sequence defined by a recurrence

Given a recurrence formula for an arithmetic sequence, $$U_{n} = \frac{1}{2+U_{n-1}}$$ Show that$$\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+ ...}}}} = (SomeGivenValue)$$ How can we solve questions ...
0
votes
2answers
45 views

Uniform convergence, wrong answer?

I have the functions $$ f_n(x) = x + x^n(1 - x)^n $$ that $\to x$ as $n \to \infty $ (pointwise convergence). Now I have to look whether the sequence converges uniformly, so I used the theorem and ...
0
votes
0answers
41 views

Convergence for trigonometric series

Does the following series converge or diverge? $$ s_{n}=\sum_{k=1}^{n}\cos\left(\frac{\pi k}{2}\right)\frac{k}{k+1000}\frac{1}{\sqrt{k}}, \text{ for n=1,2,}\ldots$$ I tried root test and ratio test, ...
1
vote
0answers
25 views

Show $\limsup S_k=y$ and $\liminf S_k=x$ from rearrangement of series

I'm in a process of proving the last part of Riemann's Theorem on conditionally convergent series. The theorem states: Let $\sum a_n$ be a conditionally convergent series with real-valued terms. ...
2
votes
1answer
47 views

Double sequence, if $(x_m)_m$ and $(y_n)_n$ converge, then they have the same limit?

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
1
vote
2answers
56 views

Double sequence, two sequences converge, but to different limits?

Let $(a_{m, n})_{m, n}$ be a "double sequence" of real numbers; that is, for every pair $(m, n) \in \mathbb{N} \times \mathbb{N}$, $a_{m, n}$ is a real number. Suppose that, for every $m$, $(a_{m, ...
1
vote
0answers
30 views

Why isn't the convergence rate 1 in ordinary differential equations?

For $${d y\over d t} = b -ay,$$ the equilibrium solution is $$y = {b\over a}$$ and the general solution is $$y = {b\over a} + k e^{-at} (k = \pm e^{c}).$$ I was asked to describe how the solutions ...
1
vote
1answer
31 views

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following

Let $f \in L^{1}([0,1])$ be a real valued function. Prove the following $1) x^k f(x) \in L^1([0,1])$ for all $k\in \mathbb{N}$ $2) \lim_{k\rightarrow\infty}\int_{0}^{1}x^k f(x) dx = 0$ $3)$ If ...
1
vote
1answer
31 views

Series generated by food donation

Not counting the fact that students need to take food from the shelves: Assume that I donate $1 $ pound of food and I get $3 $ of my students to donate, but they only donate $35\% $ of what I gave. ...
1
vote
1answer
31 views

Convergence of a sequence of Functions .

Let the function sequence $\{f_n\}$ be defined by $f_n(x)= x - 2 \exp(-nx) $ for $x \in \mathbb{R}$ . Now let $f :\mathbb{R} \rightarrow \mathbb{R} $ be defined by $f(x)= x-2I\{0\}(x)$ for $x \in ...
0
votes
4answers
86 views

Series $\sum_\limits{n=0}^\infty \frac{(n+1)}{(n^3-7)}$

I would like to prove the series $\sum_\limits{n=0}^\infty\frac{(n+1)}{(n^3-7)}$ is convergent. I have tried the ratio test but it is inconclusive, what is the way to go here ? Thanks
2
votes
1answer
25 views

Problem completing proof that the Weierstrass $\wp$ function is uniformly convergent

I'm having trouble following Ahlfors logic in his text concerning the proof that the $\wp$ function is convergent. The proof comes down to whether the series $$\sum_{\omega\neq ...
3
votes
3answers
70 views

limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$

Problem: Find with proof limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ with $a_1=1$ or show that the limit does not exist. My attempt: I have failed to determine the existence. However if the ...
-1
votes
0answers
17 views

Martingale Convergence Theorem for non-negative supermartingales: Why is limit non-negative?

Consider a supermartingale $(X_n)$ that is non-negative. The martingale convergence theorem states that $X_n \rightarrow X$ P-a.s. with $X \geq 0$ and $E[X] \leq E[X_0]$. Why can we conclude that $X ...
1
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0answers
25 views

looking for help with convergence in measure problem

Why is it true that if $(X, \mu)$ is a finite space, $f_n \to f$ in measure, and for each $n$ and $\epsilon > 0$ there exists $\delta > 0$ such that $\mu(E) < \delta \Longrightarrow \int_E ...
1
vote
0answers
52 views

Infinite sum of indicators almost sure convergence

Let $S_{n}:= \sum_{i=1}^{n} X_{i,n}$ where for each $n, X_{1,n}, X_{2,n},..., X_{n,n}$ are sequences of independents r.v.'s. $$X_{i,n}=\begin{cases}1, & \text{with probability }p_n\\0,& ...
0
votes
1answer
21 views

Find intervals where a series converges and uniformly converges

I'm not sure if I'm doing better. Here is the stuff. Consider the sequence of functions $$f_n(x) = nx \left(\frac{x}{n}\right)^n\text{sinc}^n\left(\frac{x}{n}\right)$$ and the series $$s(x) = ...
0
votes
2answers
25 views

Find the radius of convergence of this series

I used Dalamber's criteria, but when I solve the limit I find that it goes to infinity, which looks wrong. I think I might have done something wrong while simplifying the expression, but I don't quite ...
1
vote
2answers
21 views

Boundedness leading to pointwise convergence implying uniform convergence?

Consider a sequence of functions $\{f_n\}$ on some closed interval $I \subset \mathbb{R}$. Let's assume that $f_n$ is bounded on $I$ by $M \in \mathbb{R}$ for each $n \in \mathbb{N}$. If $\{f_n\}$ ...
-1
votes
0answers
28 views

What is the distribution of sum of $n$ i.i.d. exponential random variables with $n \rightarrow \infty$? [closed]

Let $X_1, X_2, \dotsc, X_n$ denote i.i.d. exponential random variables with mean $\lambda$. How does the sum $\sum_{i = 1}^{n}X_i$ converge in distribution with $n \rightarrow \infty$? I could figure ...
2
votes
1answer
28 views

Convergence in probability implies Fatou's lemma?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable ...
0
votes
0answers
22 views

What's the intuition behind this example of a power series converging everywhere on the boundary but not absolutely?

The example is $$\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for }n=\lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n$$ It seems that ...
1
vote
1answer
38 views

Number of roots of a sequence of a uniformly convergent holomorphic functions implies an upper bound for the number of roots of their limit

Let $G$ be an open, simply connected region in $\mathbb{C}$. We define a sequence of holomorphic functions $(f_n)_{n \in \mathbb{N}}, f_n: G \to \mathbb{C}$ as almost uniformly convergent iff $(f_n)$ ...
2
votes
4answers
78 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
2
votes
3answers
64 views

Prove that $\sum^{\infty}_{n=1}\frac{1}{n}\left(\frac{1+i}{\sqrt2}\right)^n$ converges but does not absolutely converge.

Prove that $\sum^{\infty}_{n=1}\frac{1}{n}\left(\frac{1+i}{\sqrt2}\right)^n$ converges but does not absolutely converge. My approach so far was to notice that ...
3
votes
3answers
49 views

Sequence convergence through epsilon proof

I need to show prove that, for any $n\in\mathbb{N}$, $$\lim_{k\to\infty} \frac{k^n}{2^k} = 0$$ using an $\epsilon$ proof. As scratchwork, I've gotten to $$|\frac{k^n}{2^k} - 0| = \frac{k^n}{2^k}\leq ...
3
votes
3answers
61 views

Show that $\int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}}$ converges.

Show that $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} $$ converges. I recognized that that since the integrand is even then $$ \int_{-\infty}^{\infty}\frac{dx}{\sqrt{x^4+1}} = ...
5
votes
3answers
50 views

Limit proof using ratio test

I have been trying to prove the limit: $$\lim_{k\to\infty} \frac{k^n}{2^k} = 0$$ for each $n\in\mathbb{N}$. I was able to do a similar proof, specifically $$\lim_{k\to\infty} \frac{k}{2^k} = 0$$ by ...
1
vote
0answers
31 views

Looking for an (outside $\Bbb R$) application of a certain theorem

I have the following theorem in the lecture notes: Let $E$ be a normed vector space and $\Omega \subset E$ be open and connected, and let $F$ be a Banach space. Let $(f_n)$ be a sequence of ...
3
votes
0answers
59 views

Prove $ \limsup a_n$ is a real number

($a_n$) is a real sequence bounded only from above. Let $S :=$ {$t \in \Bbb R:$ $t$ is the limit of a convergent subsequence of ($a_n$) }. Suppose the supremum of S is a real number. Prove that $ ...
0
votes
0answers
22 views

$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$ with $a_{2j}=(\sqrt{3})^{2j}$, different solutions

I want to calculate the radius of convergence of the series $$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$$ where $a_{2j}=(\sqrt{3})^{2j}$ and $a_{2j-1}=\frac{1}{2j-1}$. I would calculate the radius of ...
1
vote
1answer
35 views

Showing the number of converging sequences

I need to show that the following set is a finite union of convergent sequences: $$A=\frac{n-1}{n}\cos\frac{2\pi n}{3}, n\in \Bbb N$$ Can ease help me with this question? I know that cosine gets ...
-1
votes
1answer
46 views

Convergence of the method :Newton-Raphson.(GATE) [closed]

This question was asked in GATE exam and It's answer is 1. I could not understand this question. Suppose that the Newton-Raphson method is applied to the equation $2x^2+1-e^{x^2}=0$ with an initial ...
0
votes
3answers
41 views

Convergence of arithmetic mean of first $n$ numbers of a convergent sequence

Given any convergent series $(a_n)_{n \in \mathbb{N}}$,consider a new sequence $(b_n)_{n \in \mathbb{N}}$, defined as $$b_n = \frac{1}{n}(a_1 + a_2 + \dots+a_n)=\frac{1}{n}\sum_{k=1}^na_k $$ for every ...
1
vote
0answers
27 views

Existence of a solution of a limit of Fixed point equations

I am considering a setting where I am given an iid sample of symmetric positive matrices $\{S_i\}$, $i=1,\dots, n$, of a matrix valued random variable $S$ with distribution $F$. The support of $F$ is ...
0
votes
1answer
41 views

Prove the Limit Superior of a sequence is a real number

($a_n$) is a real sequence bounded from above. Let $A :=$ {$s \in \Bbb R:$ $s$ is the limit of a convergent subsequence of ($a_n$) }. Suppose the supremum of A is a real number, prove by ...
0
votes
0answers
13 views

Convergence of convolution with an even summability kernel

Suppose that $f(\theta) : [0, 2\pi] \rightarrow \mathbb{R}$ is a monotone increasing real valued function and $\{k_n\}$ is an even summability kernel. I want to show that $f \star k_n$ converges ...
1
vote
2answers
27 views

Convergence of product of (weakly) converging sequences in $L^{p}$

Preparing for an exam, I was wondering about general statements about the convergence of products. 1) Let $p, q \in ]1, \infty[$ such that $\frac{1}{p}+\frac{1}{q} = 1$ and $a_n \rightharpoonup a$ ...
0
votes
2answers
55 views

Proof of convergence

I have the following problem I want to solve with induction method. Would be great if someone helped me with it. I have $a_0=0$, $a_1=$$1\over 2$ and $|a_{n+1}-a_n|\le|a_n-a_{n-1}|^2$ I need to show ...
0
votes
1answer
49 views

Why does $ \int _1 ^3 \frac x {(x^2-9)^{4/3}}dx$ diverge?

Could someone please explain to me why the following integral is diverging? And how you would go about proving that it is. $$ \int_1^3 \frac{x}{(x^2-9)^{4/3}}dx $$
0
votes
2answers
31 views

Proving convergence for $\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^p}$

For what values of $p$ does the following integral converge: $\sum_{n=2}^{\infty} \frac{1}{n(\ln\ n)^p}.$ Ans. (Integral Test) $\int\limits_{n=2}^{n=\infty}\frac{1}{n(\ln n)^p} = \frac{1}{(-p+1)(ln\ ...
0
votes
0answers
28 views

Sequence of Functions - Pointwise and Uniform Convergence

I'm learning about sequences of functions and need some help with this problem: Show that the sequence of function $f_n(x)$ where $$f_n(x) = \begin{cases} \frac{x}{n}, & \text{if $n$ is ...
3
votes
0answers
26 views

Prove convergence in measure (i.e., in probability) “distributes” over addition and respects nonnegativity.

Suppose $X_{n}$, $Y_{n}$, and $Z_{n}$ are random variables, with $Z_{n} \geq 0$ a.s. and $X_{n} \xrightarrow{p} X$, $Y_{n} \xrightarrow{p} Y$, and $Z_{n} \xrightarrow{p} Z$. Prove the following ...
1
vote
1answer
30 views

Proving convergence

I want to prove that the following sequence is convergent: $$a_{n+1}=\frac{1}{4(1-a_n)}$$ And $a_0=0$. I should show that the sequence is increasing and bounded. I could not find a way to go about ...