Convergence of sequences and different modes of convergence.

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Convergence of series with sum

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact convex set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i ...
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34 views

How to show that the sum converges to 2? [duplicate]

$$\sum_{n=1}^{\log n} \frac{n}{2^n}$$ $n\in N$ When $\log n \to \infty$ does the sum converge to $2$? how to prove this?
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15 views

Natural numbers and convergence

If we take a sequence in the set of natural numbers, then p(x,y) must be $\geq 1$ unless x=y. Hence, we can always take $\varepsilon = 1/2$, but $p(x,y)$ is not less than $\varepsilon$ and a sequence ...
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3answers
51 views

Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques

I can't seem to use certain methods such as $\varepsilon$-N, L'Hôspital's Rule, Riemann Sums, Integral Test and Divergence Test Contrapositive or Euler's Integral Representation to prove that ...
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3answers
46 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
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54 views

Forcing series convergence

The following problem was posed by one of my lecturers: $(z_n)$ a null sequence in $\mathbb{C}$. Does there exist $(\epsilon_n)$ with each $\epsilon_n=\pm 1$ such that: $$\sum_n \epsilon_n ...
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27 views

Derivative and lipschitz

If I have a real-valued continuous function defined on a compact subset of real line, such that its derivative(wherever it exists) is bounded. Is such a function necessarily Lipschitz? Additionally, ...
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34 views

Understanding pointwise convergence vs. uniform convergence example

I'm trying to understand the difference between pointwise convergence and uniform convergence. I read this post and the last answer on it is the following: $f_n\to f$ pointwise on $(a,b)$ if for ...
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2answers
20 views

Show convergence of improper integral with nearest integer function

Suppose $f$ is decreasing and $\lim_{x\rightarrow\infty}f(x)=0$. Then why $$\int_0^\infty(-1)^{[x]}f(x)dx$$ converges? ($[x]$ is the nearest integer function). Any hint?
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2answers
60 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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94 views

Does this improper integral converge?

$$\int^\infty_0\cos x^3dx$$ I think no, because $\cos x^3$ keeps jumping between $-1$ and $1$. How to justify this rigorously?
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Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
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22 views

Show convergence of improper integral

Suppose $f(x)>0$ and $f$ is continuous on $[0,\infty)$ and $$\lim_{x\rightarrow\infty}\frac{f(x+1)}{f(x)}<1$$ How to see that $\int^\infty_0f(x)dx$ converges? I think I should use definition. ...
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1answer
18 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
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42 views

Alternating series limit question [closed]

Suppose $b_n > 0$ for all $n\geq1$ and $$\lim_{n\to\infty}n\left(\frac{b_n}{b_{n+1}}-1\right)>0,$$ show that the alternating series $\sum_{n=1}^\infty(-1)^n b_n$ converges.
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understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
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41 views

Infinite series: which test

I'm having troubling deciding which test to use for this: $$\sum_{n=1}^\infty (-1)^n\arctan\left(\frac{\ln(n!)}{n+4^n}\right)$$ I tried altnerating test and ratio test but I couldn't get an answer. ...
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15 views

How to use a base to prove something is sequentially compact.

I know this is not very specific but I'm studying for a topology exam and this is one of the things I need to know how to do. I know that part of the process is showing it converges. I was hoping ...
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2answers
34 views

Find the radius of convergence and the interval of convergence of $\sum_{n=1}^{\infty}\frac{(2n)!}{n!}x^{3n}$.

Find the radius of convergence and interval of convergence for the following summation: $$\sum_{n=1}^{\infty}\frac{(2n)!}{n!}x^{3n}$$ I don't know how to deal with the $(2n)!$ in this question. Any ...
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2answers
38 views

Explore the convergence of a series

I have to explore the convergence of a series. At this picture I used radical Cauchy indication. But I don't now what to do with a denominator to find a limit. Help me please ! Thank You so much :) ...
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32 views

Conjugate Gradient Method Near Exact Line Search

Unlike Newton-type methods, there is no natural step-length value $\alpha _k$ in conjugate gradient methods. Because of this, why do we need to use a near exact line search if we are to expect rapid ...
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1answer
67 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
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Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
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Pointwise convergence to $\ln(x)$

I came up with a stepfunction on $(0,1]$: $s_n = \sum_{i=1}^{n} \ln (\frac{i}{n}) \chi_{(\frac{i-1}{n},\frac{i}{n}]} $, where $\chi$ denotes the characteristic function. I need to show that this ...
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1answer
30 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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2answers
145 views

Convergence in probability to a non-measurable limit

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Denote the Borel field on $\mathbb{R}$ by $\mathcal{B}$. Let $\mu: \Omega \rightarrow [0,\infty)$ be a not-necessarily-measurable function and, ...
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70 views

Discuss Convergence of Series

Discuss the convergence of the given series (Please apply any test to show convergence or divergence): $$\sum_{k=1}^{\infty} \tan\left(\frac 1k\right)$$
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21 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
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45 views

Determine whether the series sin^2(1/n) converges or diverges

Determine whether the series sin^2(1/n) converges or diverges. Having real trouble with this one, I know all the terms are positive because it is being squared but I don't know where to begin with ...
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1answer
19 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
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17 views

Radius of Convergence Problem solving

I did this questions using the Ratio Test which showed that the radius of convergence is the same. I'm not sure if that is correct. (I am having my doubts about c_n becoming c_n+1 for the ratio test ...
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1answer
38 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...
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If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly

If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly. Looking at other theorems on the relationship between continuity and uniform ...
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32 views

Explore the convergence of a series with ln

How to explore the convergence of this series: $$ \sum_{n=1}^{\infty}\dfrac{1}{\ln^n(n+1)} $$ What would be better to use: De Lamber indication or feature comparison. And if comparison is a good ...
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63 views

If $\lim \limits_{n \to \infty} nb_n = 0 $, construct a convergent series such that $\lim \limits_{n \to \infty} \frac{b_n}{a_n} =0$.

The question asks to prove that if $b_n = o(\frac 1n)$ as $n \to \infty$, one can always construct a convergent series $\sum_{n=1}^{\infty} a_n$ such that $b_n = o(a_n)$ as $n \to \infty$. What I ...
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Sequence with an infinite amount of limit points

Find a sequence which has an infinite amount of limit points. I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle ...
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93 views

Prove $x_n$ converges if $x_n$ is a real sequence and $s_n=\frac{x_0+x_1+\cdots+x_n}{n+1}$

Given that $x_n$ is a real sequence, $s_n = \frac{x_0+x_1+\cdots+x_n}{n+1}$ and $s_n$ converges, $a_n = x_n-x_{n-1}$, $na_n$ converges to 0, and $x_n-s_n=\frac{1}{n+1}\sum_{i=1}^n ia_i$, prove $x_n$ ...
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50 views

Fourier Series/Parseval's Theorem

I have pretty much completed this question and have found the Fourier representation to be; $$ f(x) =\frac A2 +\sum_{n=0}^\infty 2A\frac{\cos(((2n-1)(\pi x))/2f_o)}{\pi(2n-1)} $$ Now I don't ...
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22 views

Non-linear systems: proof of quadratic convergence for Newton's method

Could someone please explain to me why r should be equal to min(R,1/(2CL)) and maybe expand a little bit on line (*) and the following?:
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31 views

Riemann zeta function and convergence-divergence

$\zeta (1.001),\zeta (1.0001)$ These are convergent. But $\zeta (1)$ is divergent. Why 1? I mean i know how to show $\zeta (1)$ is divergent. But how can we find that point. Let me rephrase that. ...
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24 views

Convergence in probability of sample variance

$X_n$ s are a sequence off iid random variables with E($X_n$) = $\mu$, V($X_n$)= $\sigma$$^2$ and $\bar X = \sum$ $\frac{X_i}{n}$. Then show that $\frac1n$ $\sum (X_i - \bar X )^2\to\sigma^2$ in ...
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Conditions for convergences of a net

I got stuck on this problem and got no clue to solve it. Can anyone one here help me? I really appreciate. Let $X$ be a set and $\mathcal{A}$ the collection of all finite subsets of $X$, ...
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22 views

Convergence of sum of binomial coefficients with negative non-integer values

I'm trying to test if the following sum converges but I have no idea how to do it: $$ \sum_{n=0}^{\infty}\binom{2\sqrt{2}-\pi}{2n} $$
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27 views

Convergence test and remainders

Let $\sum\limits_{n=1}^\infty {a_n}$ be a positive convergent sequence, whose convergence was decided by: a) Cauchy's root test. Then $R_n \leq {q^{n+1}\over 1-q}$. b) Ratio test. Then $R_n \leq ...
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1answer
18 views

Convergence and recurrence

I am asked to prove that $\sum\limits_{n=1}^\infty {\sin(n)\sin(n^2)\over n}$ converges using the following fact: Let $(a_n)_{n=1}^\infty$ be a bounded sequence. Then $\sum\limits_{n=1}^\infty ...
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61 views

A Fibonacci series

Let $F_n$ be the $n^{th}$ term of the Fibonacci sequence. That is, $F_1 = F_2 = 1$ and $F_n$ is defined recursively for $n\geq3$ by $F_n = F_{n-2}+F_{n-1}$. It is a known fact that $$ ...
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50 views

Given that $X_i$ are symmetric about 0 and iid with $E[|X_i|]=\infty$. Show that $\frac{S_n}{n}=\frac{X_1+\dots+X_n}{n}$ does not converge to 0.

I've been trying something like.. Let us assume that $\frac{S_n}{n}$ converges to 0. That means that, $$\frac{S_{n+1}}{n+1}-\frac{S_n}{n}$$ converges to 0 too. But we can rewrite this as ...
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45 views

Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $ A $ of a metric space and its limit point $ a $, we know that for every $ \epsilon > 0 $ there is another point $ x $ different from $ a $ ...
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38 views

Does it converge or diverge?

How to determine does this series converge or diverge? I have tried d'Alembert's ratio test but in the limit I get 1. I suppose I should compare it with some other series, but I can't figure out with ...
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2answers
25 views

Conditions for convergence

Let f be a continuous function on [0,1], and $f_n(x)=f(x)^n$. Under what conditions on f will the sequence converge point wise? Uniformly? I think it will converge uniformly if $-1\leq x \leq 1$ but ...