Convergence of sequences and different modes of convergence.

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Series diverges or converges

How can I show that $\sum_{n=1}^\infty (\sin n -\sin(πn/2)) / n^2$ converges or diverges? I tried using the ratio test but it's complicated I'd say it converges to $0$ since $n^2$ is growing faster ...
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1answer
365 views

Convergence of Monotone sequences? example

An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...? According to the convergence of monotone sequences if a sequences is ...
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2answers
114 views

Subspace of $C^1 [0,1]$

Consider the inner product space of continuously differentiable functions, $C^1 [0,1]$ with inner product:$$\left<f,g\right> =\int_{0}^1f(x)\overline{g(x)}\,dx + ...
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3answers
91 views

Determine convergence $\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$

Does $$\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$$ converges? Hi, I was wondering if anyone knows how to solve this problem? I think I can't use root test... because the result is ...
2
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1answer
197 views

Sequence of independent random variables: Convergence, martingales, uniform integrability

I am having some problems with the following exercise: Let $(Y_n ,n ≥ 1)$ be a sequence of independent random variables such that: $P(Y_n = e^n − 1) = e^{−n}$, $P(Y_n = −1) = 1 − e^{−n}$, $∀n ...
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1answer
38 views

Find a sequence $a_n$ such that $X_{(n)} - a_n$ converges in distribution, where $X_{(n)} = \max(X_i), i=1:n$

Let $X_1,X_2,...,X_n$ be iid and exponential(1). Define $X_{(n)} = \max(X_i), i=1:n$. What is a sequence $a_n$ such that $X_{(n)} - a_n$ converges in distribution? I think it would also converge to ...
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44 views

Handling Convergence for Derivative of a Distribution

Obtain the derivative of the distribution defined by $\rho [t] = \int_0^\infty \frac{t(x)}{\sqrt{x}}dx$, and express your answer in the form of an integral over $x$ of a formula that involves $t(x)$ ...
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1answer
180 views

Convergence of sequence in uniform and box topologies

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

Does order matter for the convergence of infinite products

Similar to infinite sums, does order matter in the convergence of infinite products? More specifically, I'm interested in the product of all rational numbers in the interval $(0,a]$. For example, ...
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3answers
81 views

find $\lim_{n \to \infty} \frac{(n^2+1)(n^2+2) \cdots (n^2+n)}{(n^2-1)(n^2-2) \cdots (n^2-n)}$

Find $$\lim_{n \to \infty} \frac{(n^2+1)(n^2+2) \cdots (n^2+n)}{(n^2-1)(n^2-2) \cdots (n^2-n)}$$ I tried to apply the squeeze theorem, yet none of my attempts led me to the solution.
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0answers
70 views

Prove $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}=3$ [duplicate]

Prove $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}}=3$ I tried to express it as a recursive sequence, but I failed.
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1answer
85 views

convergence of $\sum \frac {nx^{n-1}} {(1+x^n)^2}$

I want to examine the convergence of this function series $\sum \frac {nx^{n-1}} {(1+x^n)^2}$ for $x \in [2, +\infty)$. I showed pointwise convergence but I'm struggling with uniform convergence. I ...
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1answer
49 views

$g_n(z)=z^n$ uniformly on $D={z: |z|<1}$?

I am looking at this example: $g_n(z)=z^n$ and domain $D={z: |z|<1}$ I see that every $g_n$ will converge to 0 for $n \rightarrow \infty$. Thus it converges. Now, how can I show that it is or ...
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2answers
59 views

Testing the convergence of the given integral and finding its value

@. Does the integral $$\int_{-1}^1\sqrt{1+x\over 1-x}dx$$ exist ? If so, find it value. I compared it with $1\over \sqrt {1-x}$ and showed that its convergent. To find its values used the ...
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1answer
34 views

Closure Criterion for convergence of sequences

I know that $\{z\}=\bigcap\{\operatorname{cl}\,\{x_n\mid n\in S\} \mid S\subseteq \mathbb{N}\ \text{and}\ S\ \text{is infinite}\}$ is one of the criteria's of convergence of sequences in a metric ...
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1answer
75 views

If $b-c>1/2, b_n\to b$ and $c_n\to c$, there is $n$ such that $b_n,c_n \in (c-1/4,b+1/4)$ for all $n > N$

Suppose $b$ and $c$ are real numbers such that $b-c>1/2$. Let ${b_n}$ and ${c_n}$ converge to $b$ and $c$ respectively. Show that there exists a positive integer $N$ such that for all $n>N$, ...
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3answers
104 views

Convergence of a series

We are considering the series of general term: $(1+\frac{1}{n})^n$ I need to find if this series converges or diverges. 1) The Alembert rule can't be applied since we find the limit equal to 1. 2) I ...
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1answer
88 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
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1answer
74 views

Show convergence in probability

Let $X_1,X_2,\dots$ be I.i.d. And $S_n = X_1+X_2+\dots +X_n. $Prove if $S_n/n \to 0$ in probability then $(\max_{1\leq m \leq n}S_m)/n \to 0$ in probability. I know the idea and there is a detail I ...
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2answers
84 views

Convergence of a sequence $\frac{1}{1+n^3}$

How can I prove by integral test that the sequence $1+ \dfrac{1}{1+2^3} + \dfrac{1}{1+3^3} + \dots + \dfrac{1}{1+n^3}$ is convergent? Thank you. Is there a way that I can integrate $\dfrac{1}{1+n^3}$ ...
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1answer
39 views

Application of Stirling's theorem for the given series

I want to prove whether $x=-4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I applied alternating series test. But, while using this, I need to apply Stirling's ...
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1answer
27 views

The convergence interval of the series

I want to prove whether $x=4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I used Raabe's test. But I got limit is 1. So the test is not valid. Please help me ...
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4answers
169 views

Is this implication true? [duplicate]

Suppose that a real sequence $u_n$ is such that $$u_{n+1}-u_n \rightarrow0$$ That is not enough to prove that $u_n$ is convergent (take $u_n=ln(n)$) Now what if $u_n$ is bounded ? I guess it does ...
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1answer
27 views

If $x \in \overline{A}$ and $A \subset X$, $X$ first-countable, then there's a sequence of points in $A$ converging to $x$.

Supposedly this relies on first-countability of $X$. Let $x \in \overline{A}$, then by definition there's a neighborhood $U_1$ of $x$ that contains some $x_1 \in A$. If this is the only ...
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2answers
70 views

Test the convergence of the integral

Test the convergence of $$\int_0^\pi{\sqrt x\over \sin x}dx$$ I have to do it using comparison test. There is another test mentioned called the $\mu-test$ but its definition in the book doesn't seem ...
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118 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
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1answer
124 views

topology homework

I'm new to topology, and therefore not very good at it yet. I have following questions, that I have ansewer, please help me verify what is not correct and what is missing in my answers. Let $X$ be ...
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1answer
154 views

Question about convergence in weak operator topology (from Reed and Simon)

I am reading over Chapter VI in Simon and Reed's Functional Analysis. In the first section, the discussion covers various topologies defined on $\mathcal{L}(X,Y)$, the space of bounded linear ...
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48 views

Determine if the given integral is convergent

$$\int_0^{\pi/2}{\log x\over x^a}\,\mathrm dx,\quad a<1$$ I tried solving using the $\mu-test$. so if I consider $\mu=1$ then $\lim\limits_{x\rightarrow 0} {x\log x\over x^a}$ Solving further, I ...
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4answers
250 views

Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$

I will post the exercise below: Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ for $n \in \mathbb N$ is convergent in $\mathbb R$ with the Euclidean metric, ...
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1answer
44 views

Is there a countable Fourier transform for infinite sequences?

There's the discrete Fourier transform and the continuous one, but where's the one for infinite sequences. Let $(a_i) \subset \mathbb{C}$ be a sequence of complex numbers. The naive ways of defining ...
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0answers
218 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
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2answers
105 views

Prove that $\lim_{n\to\infty} a_k$ is nonnegative for a convergent sequence of nonnegative terms $a_k.$

Suppose we have a convergent sequence $(a_k)$ such that $a_k\ge 0 $ for all $k\ge 1.$ Show that $\lim_{n\to\infty} a_k\ge 0$. I have to prove by contradiction. This is the first time I've ...
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1answer
323 views

proof that convergence in mean implies convergence in probability

I'm attempting to understand a proof, but I am failing to see how a step is pulled off. Claim: $\text{if } f_n \longrightarrow_{L_p} f$ then $f_n \longrightarrow_{P} f$ Proof: Let $\epsilon > 0$. ...
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5answers
108 views

What is this limit equal to:

What is the following limit equal to and how do I prove it? $$\lim_{x\to 0^+} \frac{1}{1-\cos(x^2)}\cdot \sum_{n=4}^\infty{n^5x^n} $$ I've tried l'hospital but it doesn't seem to help since I don't ...
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109 views

What's convergence in a group look like?

How should we define convergence for sequences and series in groups? Here's maybe how to do it: Let $G$ be a group. A norm will be like a norm defined on a vector space except we'll define it with ...
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Find $\lim_{n\rightarrow\infty} na_n$ given that $0<a_0 <1$ and $a_{n+1}=a_n-a_n^2$ for $n\geq 0$.

I understand that no matter what value $a_0$ takes on between $0$ and $1$ that $a_1\leq \frac{1}{2}$. This has lead me to believe that for all $n\geq 1$, $b_n=(\frac{1}{n}-\frac{1}{n^2})<a_n ...
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The set of all permutations of indices such that the new series converges to the same limit forms a group?

Let $\sum_{i = 1}^{\infty} a_i = s \in \mathbb{C}$ be a convergent series of complex numbers. Then the set of all permutations $\sigma \in\operatorname{Perm}(\mathbb{N})$ such that ...
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0answers
23 views

Decomposability of a function into two

This is question on the existence of a decomposability of a function $f(v)$ into multiplicative factors $\lambda(v)$ and $g(v)$. The question is non-trivial since the functions $\lambda(v)$ and $g(v)$ ...
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5answers
209 views

Show that the sequence ${a_n}$ converges where $a_n = \sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{n}}}}$ for $n\geq 1$.

The original question was to determine whether the sequence converges, but I have checked for extremely high values of $n$ and it seems as though it does converge. This lead me to wonder if there was ...
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2answers
853 views

Absolute and Conditional Convergence of the integral $\frac{\sin(x)}{x^p}$ for real values of $p$ [duplicate]

I need to determine the values of p for which this integral converges conditionally and absolutely. $$\int_{0}^{\infty} \dfrac{\sin(x)}{x^p} dx $$ I think the interval for conditional convergence is ...
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1answer
96 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 ...
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3answers
144 views

Series Convergence

What does this series converge to? $$ \sqrt{3\sqrt {5\sqrt {3\sqrt {5\sqrt \cdots}}}} $$ and also this? $$ \sqrt{6+\sqrt {6+\sqrt {6+\sqrt {6+\sqrt \cdots}}}} $$ And, generally speaking, how ...
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2answers
146 views

Series convergence/divergence

I was trying to prove the following question. Part a is intuitive but couldn't give a clear mathematical argument. For parts b and c It seems there is something I am not seeing. Any help ? If ...
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1answer
63 views

Analyze the convergence of the series

can you please help me with this problem? I have to analyze the convergence of the following series $$ \sum_{n=1}^\infty \frac{\sin (n^{2})}{\sqrt{n^{2}+1}} $$ thanks in advance.
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1answer
166 views

how to show the convergence of an algorithm

I have two unknown variables x and y which are non linear equations to be solved. \begin{eqnarray} y=\frac {|\sin(2x+\theta)|}{\sin x\sqrt{A+2B\cos(2x+\theta)}} \nonumber \\ x=\arccos\bigg( ...
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1answer
50 views

topological sequential space $(X,\tau)$

Suppose in topological space $(X,\tau)$ every countably compact is closed.Let $(X,\tau)$ be sequential space. (1): If every infinite subset $A \subseteq X$ is closed, will $A$ be discreet in $X$? ...
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2answers
174 views

Sum of analytic functions converges uniformly but not the product

We know that if $\sum_{n=1}^\infty |f_n(z)|$ converges uniformly on $S$ to a bounded function and $f_n(z)\neq -1$ on $S$ then $\prod_{n=1}^\infty (1+f_n(z))$ converges uniformly on $S$. I want a ...
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1answer
267 views

Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
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1answer
43 views

Proving convergence using the definition.

I am studying for a test and came upon this question: Prove that $${\frac{2}{n^2}+\frac{4}{n}+3}$$ converges to $3$ using the definition of convergence. What I've been trying is ...