Convergence of sequences and different modes of convergence.

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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 ...
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4answers
96 views

why is $\lim\limits_{k\to\infty}\frac{1+k}{k^k}=0?$

My question is, why is $$\lim\limits_{k\to\infty}\frac{1+k}{k^k}=0?$$ I tried to prove it with L'Hospital's rule: $\lim\limits_{k\to\infty}\frac{1+k}{k^k}=\lim\limits_{k\to\infty}\frac{1}{kk^{k-1}}$ ...
2
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3answers
88 views

How to see this improper integral diverges?

$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$ I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
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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
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3answers
1k views

Confused by Laurent series

A typical problem related to Laurent series is this: For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
0
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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 ...
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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 ...
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1answer
12 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
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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, ...
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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, ...
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0answers
40 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, ...
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0answers
24 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. ...
4
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1answer
92 views

Proving nonexistence of sequence of continuous functions convergent pointwise to Dirichlet function (definition only)

A fellow member of the community asked: "there isn't a sequence of continuous function on $[0,1]$ that converges pointwise to the function $f$ on $[0,1]$ defined by $f(x)=0$ if $x$ is rational and ...
1
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1answer
30 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 ...
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1answer
38 views

Completeness: Nets vs. Sequences [duplicate]

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
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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 ...
3
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1answer
65 views

Prove $\{u_n\}$ converges to $\sqrt[3]{u_2^2 u_1}$ without using subsequences. [duplicate]

Prove that the sequence $\{u_n\}$ is defined by $0\lt u_1\lt u_2 \;\text {and} \;u_{n+2}=\sqrt {u_{n+1}u_n}\; \text{for}\; n\ge 1$, converges to $\sqrt[3]{u_1{u_2}^2}$. I did this using ...
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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 ...
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1answer
42 views

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

Suppose that the Newton-Raphson method is applied to the equation $2x^2+1-e^{x^2}=0$ with an initial approximation $x_0$ sufficiently close to zero. Then, for the root $x= 0$, the order of ...
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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 ...
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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
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1answer
70 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
2
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1answer
24 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
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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 ...
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0answers
50 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,& ...
10
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3answers
253 views

Is the sequence defined by the recurrence $a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$ convergent?

Let $a_0=1$, $a_1=1$ and $a_{n+2}=\frac1{a_{n+1}}+\frac1{a_n}$ for every natural number $n$. How can I prove that this sequence is convergent? I know that if it's convergent, it converges to ...
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0answers
16 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 ...
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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 ...
2
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3answers
32 views

Convergence test of $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$

Does S converge? (The answer says it converges) $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$ My attempt: Comparison test: $\ln (1+\frac{1}{k(k+2)}) \ln k \lt \ln 2 ...
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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) = ...
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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 ...
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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 $\{f_n\}$ ...
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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 $(f_n)$ ...
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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\ ...
2
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4answers
77 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
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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 ...
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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 ...
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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 ...
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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 ...
3
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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}} = ...
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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 ...
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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 ...
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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 $a_n \rightharpoonup a$ ...
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3answers
39 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 ...
0
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1answer
40 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
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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 ...
0
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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
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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 ...
0
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1answer
32 views

Pointwise and Uniform converge of $f_n(x) = \frac{nx^2}{1 + nx}, x \in [0, 1]$

I'm learning about sequences of functions and need some help with this problem: Investigate pointwise and uniform convergence of the sequence of functions $$f_n(x) = \frac{nx^2}{1 + nx}, x \in ...