Convergence of sequences and different modes of convergence.

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Leibniz test $\sum\limits_{n=1}^\infty \sin\left(\pi \sqrt{n^2+\alpha}\right)$

I am given a series: $$\sum\limits_{n=1}^\infty \sin\left(\pi \sqrt{n^2+\alpha}\right)$$ And in the description of the problem it is said that I must show by Leibniz test that it is convergent. How ...
5
votes
2answers
221 views

Prove $ \sum \frac{\cos n} { \sqrt n}$ converges

How to Prove $ \sum \frac{\cos n} { \sqrt n}$ converges Using Abel’s theorem ? I think it can be done using $\cos n = Re(e^{in})$ { Real Part of Complex Number } How to proceed ?
4
votes
3answers
116 views

Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?

I have the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$. I know that every successive partial product will necessarily be smaller than the last, as we are multiplying always by a number smaller than ...
4
votes
3answers
928 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 M>...
4
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2answers
669 views

Using the Banach Fixed Point Theorem to prove convergence of a sequence

Use the Banach fixed point theorem to show that the following sequence converges. What is the limit of this sequence? $$\left(\frac{1}{3}, \frac{1}{3+\frac{1}{3}}, \frac{1}{3+\...
4
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1answer
2k views

Weak convergence, together with convergence of norms, implies strong convergence in a Hilbert space.

Let $(x_n)$ be a weakly convergent sequence in a Hilbert space $H$. If $\| x_n \| \to \| x \|$, show that $x_n$ converges strongly to $x$. Context This problem comes from a question in my exam ...
4
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1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
4
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1answer
3k views

Prove that a given recursion sequence converges

I'm given: $$\begin{align*} x_1&=\frac32\\\\ x_{n+1}&=\frac3{4-x_n} \end{align*}$$ How do I go about to formally prove the sequence converges and show it? Thanks in advance.
3
votes
4answers
93 views

Proving convergence of a series and then finding limit [duplicate]

I need to show that $\sqrt{2}$, $\sqrt{2+\sqrt{2}}$, $\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find the limit. I started by defining the sequence by $x_1=\sqrt{2}$ and then $x_{n+1}=\sqrt{2+...
3
votes
1answer
143 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
3
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1answer
306 views

Weak closure of $\{\sqrt n e_n|n\in \mathbb N\}$ and metrizability of weak topology

Let $\{e_n|n\in \mathbb N\}$ be an orthonormal basis of Hilbert space $\mathcal H$ and put $I = \left\{\sqrt n e_n|n\in \mathbb N\right \}$. Show that $0$ belongs to the weak closure of I but no ...
3
votes
1answer
412 views

Prove that if $\sum a_k z_1^k$ converges, then $\sum a_k z^k$ also converges, for $|z|<|z_1|$

Show that if a power-series converges for any value of $z_{0}$ of $z$, it will be absolutely convergent for all values of $z$ whose representation points are within a circle which passes through $z_{0}...
3
votes
2answers
44 views

For what values does this method converge on the Lambert W function?

Someone from another question had noted that the following statement $$W(x)=\ln\left(\frac x{\ln\left(\frac{x}{\ln\left(\frac x{\ln(\dots)}\right)}\right)}\right)$$ Can be found from the identity $W(...
2
votes
3answers
129 views

Convergence of $\sum_n \frac{n!}{n^n}$

I'm working on a problem sheet and it ask to discuss the convergence of $$\sum \frac{n!}{{n}^{n}}$$ By D'Lembert's ratio test, $$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$ and so, is ...
2
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1answer
487 views

A sequence converges if and only if every subsequence converges?

I want to prove this and intuitively it makes sense. But I'm having a hard time coming up with a proof. So if a sequence converges, then we have a natural number for which the distance between all ...
2
votes
1answer
1k views

The series $\sum a_n$ is conditionally convergent. Prove that the series $\sum n^2 a_n$ 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 $ ...
1
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1answer
123 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow X$,...
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2answers
1k views

Sum of two independent random variables converges in distribution [closed]

Show that if $X_n$ and $Y_n$ are independent random variables for $1 \le n \le \infty$, $X_n \Rightarrow X_{\infty}$, and $Y_n \Rightarrow Y_{\infty}$, then $X_n + Y_n \Rightarrow X_{\infty} + Y_{\...
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3answers
213 views

How to calculate $\sum_{n=0}^\infty {(n+2)}x^{n}$

I want to calculate the sum of $$\sum_{n=0}^\infty {(n+2)}x^{n}$$ I have tried to look for a known taylor/maclaurin series to maybe integrate or differentiate...but I did not find it :| Thank you. ...
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3answers
1k views

Weak convergence and strong convergence

Suppose $f_i$ is uniformly bounded in $W^{1,p}$ for some $+\infty>p>1$, then by passing to a sub-sequence, we can suppose $f_i$ is weakly convergent to $f$ in $W^{1,p}$. Assume furthermore that $...
1
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2answers
2k views

Is $\sum\limits_{n=3}^\infty\dfrac{1}{n\log n}$ absolutely convergent, conditionally convergent or divergent?

Classify $$\sum_{n=3}^\infty \frac{1}{n\log(n)}$$ as absolutely convergent, conditionally convergent or divergent. Is it, $$\sum_{n=3}^\infty \frac{1}n$$ is a divergent $p$-series as $p=1$, and $$\...
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1answer
132 views

Convergence Problem.

Let $(a_k)$ be a sequence of real numbers and let $b_k=\frac{a_1+a_2+\dots a_k}{k}$ for each $k\in \mathbb{N}$. Prove that if $(a_k)$ converges to $\alpha\in \mathbb{R}$, then the sequence $(b_k)$ ...
0
votes
7answers
116 views

Induction and convergence of an inequality

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $. , $n\in \mathbb{N}$ My progress LHS is ...
10
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1answer
1k views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
7
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1answer
369 views

convergence of the iterated cosine

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying: $$\cos \alpha = \alpha$$ it is also evident (empirically) that ...
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3answers
402 views

Convergence of a sequence of non-negative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$.

Let $x_n$ be a sequence of the type described above. It is not monotonic in general, so boundedness won't help. So, it seems as if I should show it's Cauchy. A wrong way to do this would be as follows ...
7
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1answer
714 views

Uniform convergence of infinite series

Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets of ...
6
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1answer
70 views

Suppose $\sum\limits_{n=1}^\infty b_n$ diverges and $b_n\gt 0$, show that the series $\sum\limits_{n=1}^\infty \frac{b_n}{1+b_n}$ also diverges

As the title says, given a series $b_n > 0$, where $\sum_{n=1}^\infty b_n$ is divergent: Show that the series $$\sum_{n=1}^\infty \frac{b_n}{1+b_n}$$ is also divergent. So I've defined the series ...
6
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4answers
490 views

What steps should I be doing to determine if this series is convergent or divergent?

The problem is: $\sum_{n=1}^{\infty} \frac{1}{n(n+3)}$ The first thing I did was use the divergence test which didn't help since the result of the limit was 0. If I multiply it through, the result ...
4
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5answers
130 views

Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$

How to prove that the series $$\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$$ is convergent? What about finding the sum? My attempt: $$\ln (1-1/n^2)= \ln(n-1) -2\ln n + \ln(n+1)$$ The term in the ...
4
votes
2answers
204 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
4
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2answers
217 views

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ ...
4
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3answers
209 views

$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$ c0nvergence

$$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$$ The series converge? Please verify my solution below
4
votes
1answer
304 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
4
votes
2answers
497 views

Absolute and uniform convergence of $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$

I am trying to show $\sum _{n=1}^{n=\infty }2^{n}\sin \frac {1} {3^{n}z}$ converges absolutely for all values of $z$ $(z=0$ excepted$)$, but does not converge uniformly near $z=0$. I observed that $\...
4
votes
1answer
67 views

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$'s are disjoint. Let $f : A \to \mathbb{R}$ take elements ...
3
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2answers
122 views

Does $\sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n}$ converge?

So my teacher gave a pretty tough (for me) problem in class today: Does the series $$ \sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n} $$ converge or diverge? So far I've thought of trying the ...
3
votes
1answer
82 views

Let $A=\{\sum_{i=1}^{\infty} \frac{a_i}{5^{i}}:a_i=0,1,2,3$ or $4 \} \subset \mathbb{R}$. Then which of the following are true??

Let $$A=\bigg\{\sum_{i=1}^{\infty} \frac{a_i}{5^{i}}\ :\ a_i\in\{0,1,2,3,4\} \bigg\} \subset \mathbb{R}.$$ Then which of the following are true: a. $A$ is a finite set. b. $A$ is countably ...
3
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2answers
83 views

How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$

I'm trying to prove this sequence converges: $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ I noticed that this is continuous function which its derivative is always less than $0$ for $ x \gt 1 $, so I ...
3
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1answer
51 views

Convergence of $\int f dP_n$ to $\int f dP$ for all Lipschitz functions $f$ implies uniform integrability

I would like to prove or give a counterexample for the following statement: Let $(S,d)$ be a complete and separable space. We define: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid ...
3
votes
4answers
1k 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 ...
3
votes
2answers
647 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
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2answers
186 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ $Y(a)=\frac{1}{p}\...
2
votes
2answers
81 views

If $\lim a_n = L$, then $\lim s_n = L$

If $\lim a_n = L$, then $\lim s_n = L$, where $s_n = \frac{ a_1 + \dots + a_n}{n}$ I know that: $|s_n - L| = |\frac{a_1 + \dots + a_n}{n} - L| = \frac{1}{n}|a_1 + \dots + a_n - nL| \leq \frac{1}{n}\...
2
votes
1answer
395 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
2answers
252 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
3answers
65 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 $b_{n}=a_{n}...
2
votes
1answer
94 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
3answers
110 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
0answers
100 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... ...