Convergence of sequences and different modes of convergence.

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218 views

Convergence rate of PageRank, the problem when the second eigenvalue is complex

As far as I know the Google matrix used to calculate the PageRank is not symetric, that means that some eigenvalues can be complex, furthermore, we know that the second eigenvalue is equal to the ...
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3answers
90 views

Convergence of a series $\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$

$$\sum_{n=1}^{\infty}\frac{(n+2)!}{(3n-1)}$$ I know this series does not converge. Can someone show me how to prove that? Should i use criteria of Dalamber or any other criteria?
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1answer
42 views

Proving that a function is a contraction map

I have a function defined by: $F(X) = a \left( \frac{X-A}{\|X-A\|_2} - \frac{B-X}{\|B-X\|_2} \right) $ with $X,A,B \in \mathbb{R}^3 $ $a\in \mathbb{R}_+ $. Is this a contraction map? If yes I want to ...
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51 views

Taking limits on integration limits.

For some function $f$ and $g$ lets say that I have an integral which looks like, $\int ^{f(\epsilon)}_0 g(t,\epsilon) dt$. So if I want to compute this to zeroth order in $\epsilon$ can I just ...
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1answer
75 views

Convergence of $\frac{1}{2^k} \frac{1}{z-w_k}$

Suppose $w_1,w_2,w_3,...$ are points on the unit circle. Consider the infinite series $$\sum_{k=1}^{\infty} \frac{1}{2^k} \frac{1}{z-w_k}$$ Let $D=\{z \in \mathbb{C}: |z|<1 \}$ A) Show that series ...
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41 views

Convergence in distribution and moments

Let us assume that we are given real random variables $X_n$ that converge in distribution to $X$. Moreover, it is known that $\sup_n \mathbb{E}[g(X_n)] < \infty$, where $g$ is a measurable function ...
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1answer
24 views

What is the probability limit and limit distribution of the estimators given that$ X_i$ are iid

This is more of a practice question but I'm not sure how to really proceed. Say that $E(X)=0$ and $Var(X)=\sigma^2$. Firstly I am required to find the probability limit of the estimator as $n$ go to ...
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28 views

What is Radius of Convergence used for?

What is the applications for "Radius of convergence"? I haven't been successful in finding any information about the applications, just a lot of information about how to calculate and what it is... ...
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2answers
117 views

Proving uniform convergence of this series

I'm trying to prove that the following series $$\sum_{n=1}^{\infty}(-1)^n \dfrac{x^2 + n}{n^2}$$ converges uniformly on every finite interval $I$ in $\mathbb{R}$. In the previous exercice I've ...
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52 views

What's the most elegant way to show $x_{n+1}=\frac{2x_n^3+a}{3x_n^2}$ converges against $\sqrt[3]{a}$?

Let $1<a\in\mathbb{R}$, $x_0>\sqrt[3]{a}$ and $$\displaystyle x_{n+1}=\frac{2x_n^3+a}{3x_n^2}\;\;\;\;\;(n\in\mathbb{N}_0)$$ It's easy to show that it holds: $x_n>\sqrt[3]{a}$ ...
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25 views

Convergent Improper Integral help

I am currently studying improper integrals and came across the following problem. Analyze the convergence of the improper integral of $f(x,y) = 1 / ( x^4 + y ^2 ) $ over $R = \{(x,y) : x\geq 1, y\geq ...
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2answers
69 views

Check if $\int_1^{\infty\:}\left(e^{-\sqrt{x}}\right)dx$ converges using Convergence Test

I could use some help with an homework question: Using the convergence test, check if the following integral function converges or diverges (no need to calculate the limit itself): ...
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1answer
148 views

Does $\sum_{n\ge1}\frac1n\sin\left(\frac xn\right)$ converges uniformly?

On each bounded interval $[a,b]$ : $\left|\frac1n \sin\left(\frac xn\right)\right|\le \frac{\max\{|a|,|b|\}}{n^2}$, the series $\sum_{n\ge 1}\frac {\max\{|a|,|b|\}}{n^2}$ converges, therefore ...
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1answer
39 views

Fourier series of $f(x)$ and its convergence

sorry for the inappropriate format. I'll edit asap. question is about convergence of series. Can you explain why is "f(x)=1 ,f(x)=1/2" please.
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3answers
78 views

Proof a sequence converges to a limit

For a sequence $$a_n = \frac{\sin(n)+2}{4n^2-28}$$ How would you use the definition of a limit of a sequence to prove $a_n$ converges to $0$ I am really stuck with how this definition works, I ...
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0answers
77 views

fixed point iteration $x_{k+1}=f(x_k)$ with multiple fixed points

I have a fixed point iteration problem in my research, as below: $$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle ...
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1answer
102 views

How to find the Fourier series of $f(x)=x$?

I got a question which is too simple: Find the Fourier series of $f(x)=x$ in the interval $[-\pi,\pi]$ and show that this function doesn't converge to its Fourier series. I found the series as ...
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1answer
116 views

Solving Pell's equation: algorithm to converge $\sqrt n$

I'm trying to come up with an algorithm to solve the Diophantine equation $$ x^2 - ny^2 = 1 $$ for minimum values of $x$ when $ n $ is given. This equation is also known as Pell's Equation. The ...
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70 views

Is $\int_{1}^{\infty}\frac{dx}{\sin^{5}x}$ finite or infinite?

Is the definite integral given below finite or infinite? Why? $$\int_{1}^{\infty}\frac{dx}{\sin^{5}x}$$
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55 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
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1answer
52 views

Do one-sided limits exist for this functi0n?

Given a function $f$ with $f(x) := \begin{cases} \\ 0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}\\ 1, & \text{if }x\in\mathbb{Q}\\ \end{cases}$ , what is $\lim\limits_{x\to0^+}f(x)$ and ...
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149 views

Evaluate $\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\cdots$

Evaluate $$ \frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{4}\cdots . $$ First, it is clear that terms tend to $1$. It seems that the infinity ...
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2answers
121 views

Proving that the sequence of function isn't uniformly convergent.

Let $$f_{n}(x)=\frac{1+x}{1+\exp(nx)},\qquad n\in \mathbb{N}$$ be defined on $\mathbb{R}$. Prove that this is not uniformly convergent on the interval, that includes zero. Without proving it, ...
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1answer
37 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
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1answer
37 views

If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
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2answers
75 views

A question about convergence interval of power series

Could you give me some hint how to solve this problem: Suppose $a_n$ is sequence defined as $a_1=\frac12,a_{n+1}=\frac12\left({a_n}^2+a_n\right)$. I managed to prove that $a_n$ is decreasing ...
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97 views

What's the behavior of $\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$ outside its radius of convergence?

I want to check the behavior of $$\displaystyle\sum_{n=1}^\infty (z+\sqrt{5}+2i)^{n!}$$ outside its radius of convergence. I've tried to use the ratio test as follows: ...
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2answers
61 views

A series that converges absolutely and a bounded series

I got this exercise in my latest calculus homework assignment. I struggled with it but came up with nothing. The question is this: assume that - $\sum _{n=1}^{\infty }\:a_n$ converges ...
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3answers
83 views

Limsup Proof; Is there a more intuitive proof?

Suppose that $(a_{n}:n\geq1)$ is a sequence satisfying $\limsup_{n \to \infty}|\dfrac{a_{n+1}}{a_n}|<1$. Prove that $\sum_{n=0}^{\infty}a_n$ converges absolutly. Now the proof given is a ...
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1answer
32 views

Max of symetric random walk on Z

I can't get to solve this, could someone help me? Let $(X_i)$ be such as $P(X_i=1)=P(X_i=-1)= 0.5$ for all $i$ integer such as $1\le i\le n$. Let $S_n=X_1+...+X_n$. Let's now consider $2^n$ ...
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1answer
149 views

How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$

For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
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58 views

Laurent Series of $f(z) = \frac{1}{e^z - 1}$

The Laurent Series of $f(z)$ centred at $0$ can be written as, $$f(z) = \frac{1}{z} - \frac{1}{2} + \frac{z}{12} - \frac{z^3}{720} + \cdots$$ So we see that $f(z)$ has a simple pole at $0$. Can we ...
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1answer
138 views

Does this series converge ? $\sum_{n=1}^{\infty}\frac{\ln2\ln3\dots\ln(n+1)}{\ln(2+a)\ln(3+a)\dots \ln(n+1+a)}$

I want to evaluate $$\sum_{n=1}^{\infty}\frac{\ln2\ln3\dots\ln(n+1)}{\ln(2+a)\ln(3+a)\dots \ln(n+1+a)}$$ where $a>0$. I tried to see if it converges or not with Raabe Duhamel , but it gets really ...
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1answer
82 views

When does $ f'_{n}(x) \to g(x) =1$ imply $f'(x) =1 $

I considered the following: $f_{n}(x) \in C^1(0,1)$ (class of continuously differentiable functions) and $f_{n} \to f:(0,1) \to \bf{R} $ with $f'_{n} \to g =1$. Does this imply that $f \in C^1(0,1)$ ...
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65 views

Summability for Cauchy Product

Let $\sum_{m\in M}a_m$ and $\sum_{n\in N}b_n$ be summable. Proof that the product is summable: $\sum_{(m,n)\in M\times N}a_m b_n$ Now, let $e^{sA}:=\sum_{m\in M}\frac{s^mA^m}{m!}$ and ...
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2answers
70 views

How to find if this series is convergent or divergent

I have to find if this series is convergent or divergent. This is the series: $\sum_{n=1}^\infty{\frac{sin(5n)}{5^n}}$ I can't use the Ratio Test, and I don't know what to do with the sine in the ...
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1answer
32 views

convergence of infinite sequence

given the following sequence: $\sum_{n=1}^{\infty}\frac{1}{n\ln^{\alpha+1}{(n+1)}}$ for which values of $\alpha$ the sequence convergences? I tried to use the integral test. I defined ...
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2answers
532 views

Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $

I am revising Complex Analysis and I am a bit confused. I have a couple of results from lectures which say that $\prod_{n=1}^\infty (1+a_n)$ converges if and only if the sum $\sum_{n=1}^\infty ...
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1answer
24 views

Convergence of parametric series.

Could you give me some hint how to decide about convergence of series $\sum_{n\ge1}\left(\left(-1\right)^n+\alpha^3\right)\left(\sqrt{n+1}-\sqrt{n}\right)$, where $0<\alpha\le1$. It obvious that ...
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1answer
376 views

Infinite Series of the asymptotic expansion of Fresnel Integrals

I need to find the infinite series for the asymptotic expansions of the fresnel integrals as $x\rightarrow \infty$ and $x\rightarrow 0$. Now I have computed the asyptotic expansions to be as follows ...
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2answers
48 views

Discuss the convergence of $\int_0^1x^n \left[\log({1\over x})\right]^m \, dx$

Discuss the convergence of $$\int_0^1x^n\left[\log\left({1\over x}\right)\right]^m \, dx$$ Need some clues. I know that both $0$ and $1$ are points of discontinuities.
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1answer
58 views

Series convergence of $\frac{(-1)^n}{x^{2n+1}}$ [closed]

Does this series converge, and if so how would I prove it? I thought of using the ratio test but I'm not sure. The series is $$ \sum_{n=0}^\infty\frac{(-1)^n}{x^{2n+1}}. $$
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1answer
40 views

$\left(\frac{a_n}{n^k}\right)_n$ is bounded implies $\sum_{n=0}^\infty a_nz^n$ has a radius of convergence $\ge 1$

Let $$\left(\frac{a_n}{n^k}\right)_n\subset\mathbb{C}\;\;\;\;\;(k\in\mathbb{N})$$ be a boundet sequence. I want to show that the power series $$\sum_{n=0}^\infty a_nz^n\;\;\;\;\;(a_n,z\in\mathbb{C})$$ ...
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1answer
136 views

If a sequence of functions converges uniformly, then its limit is bounded

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of functions defined on $[a,b]$ Assume that each $f_n$ is bounded, so $|f_n| \le M_n$ for all $x \in [a,b]$. If $\{f_n\}_{n=1}^{\infty}$ converges uniformly ...
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26 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
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1answer
47 views

Convergence of a symmetric Matrix

I am kind of stuck trying to write down a mathematical proof. Since i am a Computer Sience person, I know some maths terms more and some less. There are terms which i don't know about and thus can't ...
2
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2answers
54 views

Does the series $\sum_{n\ge1}\frac{(-1)^n}{\left(n\ln\frac{n+1}n\right)^n}$ converge?

It seems to me that the general term of this series is not tends to zero : $\left(n\ln\left(1+\frac1n\right)\right)^n\sim n^n\frac1{n^n}=1$ so $\frac 1{\left(n\ln\frac{n+1}n\right)^n}\ge1$. Am I ...
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1answer
44 views

Convergence of a Power series

Consider the power series $\sum^{\infty}_{n=0} a_nx^n$. It is fairly easy to impose conditions on the value of $x$, so as to make the series convergent. However, I was wondering if it is possible to ...
0
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1answer
19 views

What convergence test can I use on this series?

I am doing practice problems for an exam, and I am not sure how to test this series: Limit from n=1 to infinity of cos(n) * sin^2(1/n) I am supposed to use lim x -> 0 sin(x)/x = 1 to find the ...
0
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3answers
59 views

Determining the convergence of $\int_{1}^{2}\frac{1}{\sqrt{x}(2-x)}\, \mathrm{d}x$ in simple way.

Is there a (simple) way to determine its convergence without determining its value? I know that the function $x\mapsto \left (\sqrt{x}(2-x) \right )^{-1}$ is continuous for all $x\in ...