Convergence of sequences and different modes of convergence.

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567 views

If $\sum a_n^2 n^2$ converges then $\sum |a_n|$ converges

Let's suppose that $(a_n)$ is a sequence so that $\sum a_n^2 n^2$ converges, so I have to prove that $\sum |a_n|$ converges. By the Cauchy criterion we have that, since $\sum a_n^2 n^2 $ converges ...
2
votes
0answers
92 views

Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$

This is by far my most ambitious proof attempt to date and I'm not very good at them; so even if the proof is correct I would still appreciate feedback on nomenclature, clarity, elegance, etc... ...
2
votes
3answers
98 views

$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

how can i show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
2
votes
3answers
61 views

$a_n$ is bounded and decreasing

my second question from [An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$ Let $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and ...
2
votes
1answer
69 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
2
votes
1answer
65 views

Measure Spaces: Uniform & Integral Convergence

Given a measure space $\Omega$. Consider a sequence of measurable functions $f_n$ Suppose it converges pointwise: $f_n\to f$ Can one find increasing subsets with uniform convergence: ...
2
votes
4answers
179 views

Convergence of averaged sine function

I have stumbled upon those two problems which I got a little stuck on that is show convergence or divergence for the series $$\sum_{n=1}^{+\infty}\frac{\cos(n)}{n}$$ and ...
2
votes
2answers
139 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 $ ...
2
votes
1answer
413 views

The series $a_n$ is conditionally convergent. Prove that the series $n^2 a_n$ is divergent.

Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $. Can I use limits? We know: $\lim\limits_{n \to \infty} a_n = 0 $ ...
2
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3answers
2k views

Proof that rational sequence converges to irrational number

Let $a>0$ be a real number and consider the sequence $x_{n+1}=(x_n^2+a)/2x_n$. I have already shown that this sequence is monotonic decreasing and thus convergent, now I have to show that $(\lim ...
2
votes
1answer
213 views

A problem about convergence…

I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
2
votes
3answers
534 views

Boundedness and pointwise convergence imply weak convergence in $\ell^p$

Let $p\in(1,+\infty)$ and consider the space $\ell^p$ with its usual norm. The following are equivalent: (1) $x_n \rightharpoonup x$ (i.e. $x_n$ weakly converges to $x$); (2) $$\exists ...
2
votes
3answers
378 views

$L^p$-space convergence

Let $({f_n})_{n\geq 1}$ be a sequence of functions on $[0,1]$ such that $\lim _{n\rightarrow \infty}f_{n}(x)=f(x)$ almost everywhere with respect to Lebesgue measure and $$\sup_{n\geq ...
2
votes
4answers
842 views

p-series convergence

Show that if $p>1$, $\sum\frac{1}{n^{p}}$ converges and if $p<1$ it diverges for $p\in\mathbb{R}^{+}$. Is there any way to show another series converges or diverges and then use the Comparison ...
2
votes
2answers
238 views

Prove that the sequence converges

Prove that the sequence converges. For each positive integer $n$, let $$y_n = 1 + \frac12 + \frac13 + \cdots + \frac1n - \int_1^n \frac{dx}x.$$
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vote
6answers
153 views

Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
1
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4answers
55 views

If $x_n \to a$ and $x'_n \to a$, then $\{x_1, x'_1, x_2, x'_2, …\} \to a$

I know that if all subsequences of $\{x_1, x'_1, x_2, x'_2, ...\}$ converge to $a$, then $\{x_1, x'_1, x_2, x'_2, ...\}$ converges to $a$, but I only know two subsequences of $\{x_1, x'_1, x_2, x'_2, ...
1
vote
1answer
45 views

Error Analysis and Modes of Convergences

I have the following question regarding Modes of Convergence Recall that the Taylor series of the function $f(x) = \ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$. Let ...
1
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3answers
53 views

Dominated Convergence: Estimate

This is an application of: Spectral Measures: Domain Criterion I'm trying to check the estimate: $$\frac{1}{h}|e^{ixh}-1|\leq C\left(|ix|+1\right)\quad(h\in(-\varepsilon,\varepsilon))$$ for some ...
1
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1answer
62 views

Investigating the convergence of a series using the comparison limit test

Actually not sure how to approach this... but I may be missing something: Replacing the sequence: $x_{n}=1+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}-2\sqrt{n},\,\,\,\, n=1,2,....$ By the ...
1
vote
1answer
90 views

Prove that $r^n/n!$ converges where $n\ge r$ [closed]

The answer is in the title of the question. I need to show it converges to 0 and $r>0$. I am sorry if this is a bad question, I'm having trouble explaining it. So essentially this Do the ...
1
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1answer
67 views

Range of convergence for Taylor's series (about 0) for e^(sin x)

Is there anything wrong with my method below? Also, is there an easier method? For $sin\,x = \sum^{\infty}_{k=0}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$, $L_1 = \lim_{k\rightarrow\infty} \left| ...
1
vote
2answers
51 views

Not sure which test to use?

Trying to determine if the following series is convergent: $$\sum_{k = 1}^{\infty} {2^k ln(1+1/(3^k))}$$ I have no idea how to compute the integral so im not sure if I should use the integral test, ...
1
vote
2answers
61 views

Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $

I'm stuck on a homework question, and could really use some help. Here is said question: "Assume that for every $n$ the following occurs: $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ when $ 0<q<1 $ ...
1
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1answer
357 views

Prove that the sequence of L-Lipschitz functions converge

$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz: means - $|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions) and assume $f_n \to f$ in a pointwise ...
1
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1answer
102 views

‎sequential ‎space

A‎‎ ‎sequential ‎space ‎has ‎unique ‎sequential ‎limits ‎iff ‎each ‎countably ‎compact ‎subset ‎is ‎closed. ‎Proof: ‎If‎ ‎$ \{ x_n \} $ ‎is a‎ ‎sequence ‎converging ‎to ‎two ‎distinc ‎‎$‎x‎$ ‎and ...
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1answer
32 views

A hereditarily Lindelöf $KC$-space $( X,τ )$ is Katětov-$KC$ if and only if there is a weaker sequential $US$ topology $σ⊂τ

A space $( X,τ )$ is said to be Katětov $ KC $ if there is a topology $ σ⊂τ$ such that $( X,σ )$ is minimal $ KC $. The notion of strongly KC-spaces, that is, those spaces in which every ...
1
vote
2answers
75 views

Must the sequence $X_n$ converge to $0$ in probability?

Let $X_1, X_2,\dots$ be a sequence of random variables with $\lim_{n\to +\infty} E[|X_n|] = 0$. Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
1
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1answer
634 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 ...
0
votes
2answers
71 views

Convex set weakly closed if and only if strongly closed as well

I'm looking for a proof that given $(X\textbf{ } \|\cdot\|)$ normed space, $M \subset X$ convex set, $M$ is weakly closed if and only if it's strongly closed as well.
0
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0answers
90 views

Semigroups: Entire Elements (II)

Problem Given a Banach space $E$. Consider a contraction C0-group: $$T:\mathbb{R}\to\mathcal{B}(E):\quad\|T(t)x\|\leq\|x\|$$ Define its generator by: $$Ax:=\lim_{h\to0}\frac{1}{h}(T(h)x-x)\in E$$ ...
0
votes
1answer
87 views

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 ...
0
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2answers
81 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) = ...
0
votes
3answers
88 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!
0
votes
1answer
303 views

Convergence of sequence in uniform and box topologies

I am trying the following problem: $w_1=(1,1,1,1,\ldots)$ $w_2=(0,2,2,\ldots)$ $w_3=(0,0,3,3,\ldots)$ $\cdots$ $x_1=(1,1,1,1,\ldots)$ $x_2=(0,\frac{1}{2},\frac{1}{2},\frac{1}{2}\ldots)$ ...
0
votes
2answers
94 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 ...
0
votes
2answers
52 views

Positive limit of sequence vs. positive terms

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
41 views

Series Summation,Convergence

I am stuck on the 4 th one.I have done the rest.I have found out the value of a_n.But not getting how to proceed for the 4 th one.
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1answer
72 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
190 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 ...
23
votes
2answers
765 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 $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
5
votes
2answers
251 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 ...
16
votes
3answers
963 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 ...
15
votes
1answer
274 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: ...
36
votes
0answers
2k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
10
votes
1answer
229 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 ...
13
votes
2answers
557 views

Infinite sum of reciprocals of pentagonal numbers

How do I find this sum: $$\sum_{n=1}^\infty \frac{1}{p(n)}$$ where $p(n)=\dfrac{n(3n-1)}{2}$ is the $n$th pentagonal number? I know it is a convergent series, but I don't know if the sum can be ...
12
votes
2answers
554 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 ...
10
votes
2answers
218 views

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
9
votes
1answer
226 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 ...