Convergence of sequences and different modes of convergence.

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Prove or disprove regarding sequences

Question: Find a sequence $\displaystyle \{a_n\}_{n=1}^{\infty}$ such that $a_n\rightarrow 0$ and $n\left|a_{n+1}-a_n\right|\rightarrow \infty$. If no such sequence exists, prove it. My try: At ...
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3answers
71 views

Does $\{(1-x)x^k\}$ converge uniformly on $[0,1]?$

The convergence is clearly pointwise to the function $f(x) = 0$, but I'm not sure if this convergence is uniform. I wanted to prove that it wasn't uniform because I had a feeling it wasn't, but I'm ...
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1answer
50 views

Weak convergence in $l_p$ implies pointwise convergence?

Could someone please share their thoughts on this one: Consider at $l_p(Y)$, for $1<p<\infty$ with the counting measure on $Y$. Show that if a sequence weakly converges in $l_p(Y)$ then it ...
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28 views

composition of uniformly convergence sequence with continuous function, is uniformly convergence?

Let $(f_n)$ be a series of functions in $C[0,1]$ that uniformly converge to a continuous function $f\in C[0,1]$. a. Let $g: [0,1]\to [0,1]$ be a continuous function. Is it true that $f_n\circ g$ ...
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1answer
29 views

Complete convergence and almost sure convergence of random variables

Let $X_{n}$ be a sequence of independent random variables. Prove that $X_{n}$ converges to zero, almost everywhere (a.e.) if and only if for all $\epsilon >0$, $\sum_{n=1}^{\infty } ...
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38 views

Convergence of Expectations

Suppose $\{X_n\}$ is a sequence of non-negative random variables such that $$EX_n<\infty, \text{ }\lim_{n\rightarrow \infty}EX_n =\infty$$ and $\lim_{n\rightarrow \infty}X_n$ exists a.s. May I ...
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12 views

Numerical analysis: prove that rate convergence p is the limit of log |en+1|/ log |en|

For the iterative method of $ X_{n+1} = g(X_n) $ with convergence rate of p and $$ \lim_{n\to \infty} \frac{ |e_{n+1}|}{|e_n|^{p} } = C $$ I need to prove that: $$ \lim_{n\to \infty} \frac{ log ...
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30 views

For What Value of P is $\sum (-1)^{n-1}\frac{(\ln(n))^p}{n}$ convergent?

The Question for what value of p is $\sum (-1)^{n-1}\frac{(\ln(n))^p}{n}$ convergent? My Work It's an alternating series, so I have to show which p values it's limit to infinity go to $0$ and I ...
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1answer
20 views

Show that $\sum(-1)^{n+1}ne^{-n}$ is convergent

The Question Determine if $\sum(-1)^{n+1}ne^{-n}$ converges or diverges My Work It's an alternating series, so if it meets its conditions, I could use the alternating series test. I took the ...
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1answer
29 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
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Derivations: Analytic Vectors

Given a C*-algebra with unit $1\in\mathcal{A}$. Consider a dynamics $\tau$. Define its derivative: $$\delta(A):=\lim_{t\to0}\frac{1}{t}\left(\tau^t(A)-A\right)\quad A\in\mathcal{D}(\delta)$$ (The ...
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3answers
80 views

Convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}\Big(\sqrt[3]{(n+1)^{2}} - \sqrt[3]{n^{2}}\Big)$

$\displaystyle\sum_{n=1}^{\infty}\frac{\sqrt[3]{(n+1)^{2}} - \sqrt[3]{n^{2}}}{n}$ Converging or Diverging? I guess I have to lower the fraction so that the roots will get away and I will have ...
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2answers
14 views

Using Weistrass Approximation Theorem to define fourier series convergence.

Weistrass Approximation Theorem: Let f be continuous on [-$\pi$,$\pi$] with $f(-\pi)=f(\pi)$. Then for each $\epsilon>0$ there is a trigonometric polynomial T such that $|f(x)-T(x)|<\epsilon$ ...
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1answer
42 views

Topology: Convergent subsequence implies compactness

I've looked at the related questions/answers to my problems, but I need clarification. I don't understand the contradiction from the following proof. : I want to show that in the metric space ...
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19 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on ...
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2answers
35 views

Test Convergence of the Series [duplicate]

Is $$\sum^\infty_{n=1}\frac{2n^2}{5n^2+2n+1}$$ Convergent or Divergent? If convergent, find the sum. If Divergent, explain why. Since it is continuous, positive, and decreasing I used the ...
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1answer
22 views

Determine whether the series is convergent or divergent.

I am confused as to what test I should use to determine whether it's convergent or divergent. $\sum_{n=1}^\infty \frac{n+5^{}}{{(n^{7}+n^{2})}^{1/3}}$
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2answers
50 views

Is the Series Convergent or Divergent?

Is $$\sum^\infty_{n=4}\frac{3^{2n}}{(-10)^n}$$ Convergent or Divergent? Explain why. I know I can do: $$\sum^\infty_{n=4}\frac{9^{n}}{(-10)^n} \Rightarrow ...
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126 views

Convergent or Divergent Sequence?

Is $$\frac{3n + (-1)^n}{n^6 + 5n}$$ Convergent or Divergent? Explain. For the limit as $n\rightarrow\infty$ I got the following: $$\lim_{n\rightarrow\infty}\frac{3n + (-1)^n}{n^6 + ...
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2answers
36 views

Why is the improper integral $\int^0_{-\infty}{1\over 3-4x}dx$ divergent?

Take the following: $$\int^0_{-\infty}{1\over 3-4x}dx$$ Substituting $t$ for $-\infty$, we can replace the above with $$\left.\lim_{t\to-\infty}-{1\over4}\ln(3-4x)\right\rbrack^0_{t}\ ,$$ which ...
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20 views

Product of convergent series that diverges

Can the product of two convergent series diverge if one of the series is absolutely convergent, and one is not? (Here, product means product component-wise). I can't think of counterexample.
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67 views

Determine whether the series $\displaystyle\sum_{n=0}^\infty\frac{2^{n^2}}{n!}$ is convergent or divergent.

Determine whether the series $$\sum_{n=0}^\infty\frac{2^{n^2}}{n!},$$ is convergent or divergent. I know I have to use the ratio test.
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If $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$

This is the $1988$ Putnam $B4$ Problem: Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty a_n^{n/({n+1})}$. My problem lies in ...
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1answer
48 views

If $\sum_n a_n $ converges and $a_n > 0$, is it true that $\sum_n \sqrt{a_n}$ is also convergent?

Assume that $\sum_n a_n $ is convergent and $a_n > 0$, is it true that $\sum_n \sqrt{a_n}$ is also convergent? What can you comment on $∑_n a_n^2$? I have thought that since summation an is ...
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1answer
53 views

Weak Law of Large Numbers for a non-iid, non-ergodic sequence

I have a somewhat open-ended question. Let's say I have a sequence of random variables $(X_n: n \geq 1)$ which are neither independent, ergodic, nor identically distributed. Normally I would say that ...
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2answers
22 views

Prove that : $\lim_{n\to \infty}a_n = L \iff\lim_{n\to \infty}b_n =L $

let $a_n$ be a sequence. we define $b_n=a_{n+k}$, $k\in \mathbb {N}$, prove that $\lim_{n\to \infty}a_n = L \iff \lim_{n\to \infty}b_n =L $ I need to formally prove this, can someone give me a ...
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45 views

$\displaystyle\sum_{n=1}^{+\infty}\arctan(n)/n$ converges or diverges?

Try ration test but fail. I have no idea about that. I know that $0<\arctan n < \pi/2$ but $(\pi/2)/n$ is divergent. I try on wolframalpha and it say the series is conditionally converge ...
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43 views

Abel's test and Leibnitz's test

Hi does anyone know how Abel's test and Leibnitz's test( also called the 'alternating series test' for convergence) are related? Is the alternating series test sometimes called Abel's test? The ...
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82 views

Convergence of the sequence $\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \dots+\frac{1}{\sqrt{n^2+n}}$

How to determine the convergence of this sequence? $$\alpha_n = \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2+2}}+ \dots +\frac{1}{\sqrt{n^2+n}}$$ I was trying to show first that the sequence has a ...
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1answer
22 views

Convergence of Power Series $\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$ with $\alpha ,\beta \ge0$

For $\alpha, \beta \geq 0 \in \mathbb{R}$, find the radius of convergence for the series: $$\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$$ Ok, so if $\alpha$ and $\beta$ are $\leq ...
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18 views

Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge.

Find a series $f(r)=\sum_{0}^{\infty}a_nr^n$ s.t converges for $|r| < R$ and s.t. $\lim_{r\rightarrow R-} f(r)$ exists but series does not converge. How do I approach this? I think that $a_n r^n$ ...
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1answer
38 views

Show that operator T is a contraction mapping

I want to check whether the operator T defined as: $Tf(x) = \beta \left\lbrace \sum_{\theta} \mu_\theta \left[ h_\theta(x) + \int f(x') Q_\theta(x,dx') \right]^\alpha \right\rbrace^{1/\alpha} $ is a ...
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26 views

Convergence in $L_p$ and elsewhere

Let $\|f\|_p:=(\int_X|f|^pd\mu)^{1/p}$ and let $L_p$ be the space of (the classes of equivalence of) complex or real measurable functions such that $\int_X|f|^p d\mu<\infty$ exists. In ...
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Method for solving radius of convergence problem

Hi I am interested if the following method for solving for the radius of convergence for power series problem is a valid method: Find the radius of convergence of the following: ...
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28 views

Sequence that converges as for the norm but not almost everywhere

How can I find a sequence that converges as for the norm but doesn't converge almost everywhere, in some space $L^p$ ?? Could you give me some hints ??
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24 views

Method of finding radius of convergence

Hi is it acceptable to evaluate the radius of convergence $R$ of this power series $$\sum_{n=1}^{\infty}(-1)^{n}n^{-\frac{2}{3}}x^{n}$$ by instead of taking $a_{n} := (-1)^{n}n^{-\frac{2}{3}}$ we take ...
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vector bundle and convergence of operators

Let $ (E, B, π, V)$ a fiber bundle. Suppose that $x_n \rightarrow x$ with $(x_n), x\in B$ then by the local triviality condition $\phi(x_n,.):V \rightarrow E_{x_n}=\pi^{-1}(x_n)$ is an isomorphism for ...
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question about using iteration to solve equation

what is the reason for rapid convergence using iteration process to solve an equation? When I was solving an equation using iteration process, I used to possibilities to re-write a function so came ...
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40 views

Checking for convergence/divergence of the nth root.

I've been working on this problem for a while, but I can't really get it. I get it, but I don't actually get it. The question is to find whether or not this series converges: ...
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1answer
27 views

Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

! (http://i.imgur.com/Zwt1m1n.png) I need to do the question at the top of this image. I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim ...
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Complex Power Series Convergence and Expansion

Can we have a power series of the form $\sum_{k=0}^\infty c_k(z-1)^k$ which converges at $z = 3$ but diverges at $z = 0$? This is for homework and I just don't know where to start. Tips would be ...
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Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
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How to determine the convergence of series $\sum_{n=1}^\infty \frac {2^{-n}}n$ and $\sum_{n=1}^\infty \frac n{n^2+3n+5} $? [closed]

the series $$\sum_{n=1}^\infty \frac {2^{-n}}n$$ the series $$\sum_{n=1}^\infty \frac n{n^2+3n+5} $$ How can we find their convergence?
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61 views

Limit of a finite sum

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence defined by $$a_n = \frac{n+1}{2^{n+1}}\left(\sum_{k=1}^n \frac{2^k}{k}\right)$$ Show that the sequence converges and find its limit. Update: After some ...
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54 views

prove that $\lim \limits_{n \to \infty}a_n= \lim \limits_{n \to \infty}b_n$

$\{a_n\}$ and $\{b_n\}$ are two converging sequences. Its given that the two sets : $\lbrace n \in \mathbb{N} : a_n \le b_n \rbrace$ , $\lbrace n \in \mathbb{N} : b_n \le a_n\rbrace$ are not ...
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59 views

Check if the sequence is convergent?

Check if the sequence $a_n=\sqrt[n]{\sum_{k=1}^{n}(2-\frac{1}{k})^k}$ is convergent? How to do this?
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6answers
216 views

Check if the sequence is convergent

Check if the sequence $$\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}$$ is convergent. I really don't know how to start.
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1answer
26 views

Convergence of a sequence of functions integrated over a sequence of measures

I have real-valued functions $\{f_n\},f$ on a subset $X\subset \mathbb R^n$ that are equicontinuous and I have Borel measures $\{\mu_n\},\mu$. I have that For each fixed $m$, $\int f_m d\mu_n\to\int ...
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2answers
27 views

Example of two sequences that have opposite relation between their values and limits.

Is there an example of two sequences $a_n$ , $b_n$ that fulfill the following conditions? $a_n$ and $b_n$ are not constant sequences. $\lim_{n\rightarrow\infty}b_n$ $<$ ...
2
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0answers
36 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...