Convergence of sequences and different modes of convergence.

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1answer
561 views

Proof of convergence in distribution of a discrete random variable

I'm working on a question on "convergence in distribution" and I appreciate if you could guide me on how to approach this question: Here is the question: Let $X_n$ be integer-valued random ...
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33 views

Semigroups & Generators: Entire Elements: Construction

Problem Given a Banach space $E$. Consider a $\mathcal{C}_0$-group(!): $T:\mathbb{R}\to\mathcal{B}(E)$. Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)$$ (The domain being those ...
0
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2answers
66 views

Taylor series expansion and the radius of convergence

Hello I have some problems concerning Taylor series. Given the function $$f(x)=e^{\sin{x}} $$ I concluded that the Taylor series expansion would be $$f(x) = ...
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3answers
84 views

How to show that $\sum {2^j + j \over 3^j - j}$ converges

I'm not quite sure how to go about growing the numerator or shrinking the denominator to perform a tricky comparison test, or hashing out the ratio test, so any help would be much appreciated!
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2answers
88 views

$ KC $ spaces imply $ US $ spaces , but vise versa is false.

In the $ US $ space , each convergent sequence has unique limit. In the $ KC $ space , every compact subset is closed. It easy to show that $ KC $ spaces imply $ US $ spaces. The ...
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2answers
50 views

real analysis:convergence

Let $\{x_m\}$ be a sequence in $E_1$ that converges to $L \in E_1$. a. prove that if $L>0$ and there exists $n \in N$ such that for all $m >n$ holds that $x_m > 0$ b. True or false? If for ...
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1answer
71 views

Alternating functional Series Convergence SOS…

Does the following series converge? $\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$ what is the radius of convergence?!!
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0answers
181 views

Fibonacci Numbers - Complex Analysis [duplicate]

Possible Duplicate: Complex Analysis - Integral over a circle of radius R Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
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2answers
726 views

How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
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2answers
241 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
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1answer
248 views

Does $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ converge for some $a\in(0,1)$?

The divergence of the series $\sum_{n=1}^\infty n^{-1-|\sin n|}$ is proved here. An inmediate consequence is that if $a\ge1$ then $\sum_{n=1}^\infty n^{-1-|\sin n|^a}$ also diverges. My question is: ...
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3answers
884 views

Convergence\Divergence of $\sum\limits_{n=1}^{\infty}\frac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$

Prove convergence\divergence of the series: $$\sum_{n=1}^{\infty}\dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}$$ Here is what I have at the moment: Method I My first way uses a result that ...
9
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1answer
223 views

Prove the divergence of a particular series, given that another series diverges

Suppose $\{a_i\}_{i\in\mathbb N}$ is an increasing sequence of positive real numbers such that $$\sum_{n=1}^\infty\frac{1}{a_n}=+\infty.\tag{1}$$ Then I have to show that also ...
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2answers
525 views

Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$ \sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}, $$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
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1answer
732 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...
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1answer
1k views

Common Problems while showing Uniform Convergence of functions

All, I am having some trouble in checking whether a sequence of functions converges pointwise or uniformly. The problem is, sometimes, my intuition is right and sometimes its wrong. Finding the limit ...
13
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3answers
257 views

Find the limit of $(\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot … \cdot \sin 1)^{\frac{1}{n}}$

Could you tell me how to find $\lim_{n \rightarrow \infty} (\sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot ... \cdot \sin 1)^{\frac{1}{n}}$ ?
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0answers
167 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
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3answers
613 views

Why can't $\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}>2$?

So we have$$\sqrt{2}^{\sqrt{2}{^\sqrt{2}{^\cdots}}}=x\\\sqrt{2}^x=x$$where $x=2$ heuristically seems like a good solution. However, $x=4$ seems like an equally good solution. I was told in passing ...
18
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2answers
948 views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...
12
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0answers
227 views

Showing a series is convergent. [duplicate]

Possible Duplicate: Contest problem about convergent series Let ${p}_{n}\in \mathbb{R} $ be positive for every $n$ and $\sum_{n=1}^{∞}\cfrac{1}{{p}_{n}}$ converges, How do I show that ...
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2answers
2k views

Does convergence in distribution implies convergence of expectation?

If we have a sequence of random variables $X_1,X_2,\ldots,X_n$ converges in distribution to $X$, i.e. $X_n \rightarrow_d X$, then is $$ \lim_{n \to \infty} E(X_n) = E(X) $$ correct? I know that ...
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2answers
511 views

Prove $\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty$ for an increasing sequence $a_n$ of positive integers

The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that: $$ \sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty $$ While I have ...
9
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1answer
317 views

Sum of cosines of primes

Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$ How to prove this series converges/diverges? $$\sum_{n=1}^\infty \cos{p_n}$$
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2answers
430 views

convergence of $\sum \limits_{n=1}^{\infty }\left\{ \dfrac {1\cdot3 \cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2}$

I am investigating the convergence of $$\begin{split}\sum _{n=1}^{\infty }\left\{ \dfrac {1\cdot 3\cdots (2n-1)} {2\cdot 4\cdots (2n)}\cdot \dfrac {4n+3} {2n+2}\right\} ^{2} &= \sum _{n=1}^{\infty ...
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2answers
482 views

Convergence/Divergence of a particular infinite nested radical

Is it known if the following infinite nested radical converges or diverges (for $n \in \mathbb N$)?: $$R(n) = \sqrt{n+\sqrt{(n+1)+\sqrt{(n+2)+ \cdots}}}$$ I recently became interested in these ...
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2answers
2k views

Dini's Theorem. Uniform convergence and Bolzano Weierstrass.

In Spivak's chapter on uniform convergence he asks to prove the following THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...
4
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1answer
391 views

Dominated convergence and $\sigma$-finiteness

I am curious about the Dominated Convergence Theorem for a sequence of functions that converges in measure. Theorem: Let $(X,\mathcal{S},\mu)$ be a measure space. If $\{f_n\}, f$ are measurable, ...
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3answers
320 views

How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]

Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.
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1k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
14
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1answer
340 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
14
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2answers
872 views

If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?

This is a self-posed question, so I do not know the answer and I would like to know what do you think about. Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
13
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1answer
492 views

A few counterexamples in the convergence of functions

I'm studying the various types of convergence for sequences of real valued functions defined on measure spaces: pointwise convergence a.e. , convergence in $L^p$ norm, weak convergence in $L^p$, and ...
12
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2answers
326 views

Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?

Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
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2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
8
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2answers
538 views

Prove that the convergence of the sequence ($s_n$) implies the convergence of ($s_n^3$)

I believe I have the gist of how to prove this. My professor worked out a problem similar to this one only, instead of ($s_n^3$), he used ($s_n^2$), and I am slightly confused as to how he came up ...
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194 views

Again, improper integrals involving $\ln(1+x^2)$

How can I check for which values of $\alpha $ this integral $$\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}\,dx $$ converges? I managed to do this in $0$ because I know $\ln(1+x)\sim x$ near 0. I ...
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2answers
1k views

Does $\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $ converge/diverge?

How would you prove convergence/divergence of the following series? $$\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $$ I'm interested in more ways of proving convergence/divergence for this series. Thanks. ...
6
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0answers
108 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
6
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3answers
332 views

Application of Central Limit Theorem

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
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3answers
3k views

Lebesgue Dominated Convergence example

How do I use Lebesgue Dominated Convergence Theorem to evaluate $$\lim_{n \to \infty}\int_{[0,1]}\frac{n\sin(x)}{1+n^2\sqrt x}dx$$ What dominating function to use here?
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1answer
241 views

Is this infinite series related to prime and composite numbers convergent?

I don't know whether this series converges: $$(\frac{1}{4} - \frac{1}{5}) + (\frac{1}{6} - \frac{1}{7}) + (\frac{1}{8} + \frac{1}{9} + \frac{1}{10} - \frac{3}{11}) + (\frac{1}{12} - \frac{1}{13}) + ...
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1answer
52 views

The interchange of limit for integration

This kind of problem bothers me for a while. Each time I meet such problem I got stuck and has to deal them case by case. So I post this problem here to ask for some general condition of the ...
2
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2answers
105 views

Find the Maclaurin series of f(x)

Find the Maclaurin series of $f(x)=\dfrac{1}{1+x+x^2}$ and the radius of convergence of the series. I can't solve this problem.
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1answer
2k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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2answers
108 views

trouble calculating sum of the series $ \sum\left(\frac{n^2}{2^n}\right) $

Find out the sum of the series $\displaystyle \sum\limits_{n=1}^{\infty} \dfrac{n^2}{ 2^n}$. I have checked the convergence, but how to calculate the sum?
16
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1answer
605 views

Series which are not Fourier Series

How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function? I have shown that the series is convergent by Dirichlet test. Let $a(n)=\frac{1}{\log ...
13
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2answers
194 views

Convergence of power towers

Let's define the sequence $\{s_n\}$ recursively as $$s_1=\sqrt2,\ \ \ s_{n+1}=\sqrt2^{\,s_n}.$$ Or, in other words, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
9
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3answers
2k views

In what spaces does the Bolzano-Weierstrass theorem hold?

The Bolzano-Weierstrass theorem says that every bounded sequence in $\Bbb R^n$ contains a convergent subsequence. The proof in Wikipedia evidently doesn't go through for an infinite-dimensional space, ...
8
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1answer
833 views

Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...