Convergence of sequences and different modes of convergence.

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2
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4answers
79 views

I can't figure out how to get this sequence to converge to 0

I'm trying to prove this sequence converges to $0$ (or maybe I'm wrong..): $$k\in\mathbb N,1<q,\lim_{n\rightarrow\infty}\frac{n^k}{q^n}$$ I'd be happy to get some help with this! thank you!
7
votes
3answers
215 views

How can I prove the convergence of a power-tower? [duplicate]

In here, I saw that $$x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$$ exists as a real number (convergent) if and only if $$x\in[e^{-e}, e^\frac{1}{e}].$$ How can I prove this??
4
votes
3answers
86 views

Convergence of $\int_0^\infty $sin$ (x^p) dx$

Consider the $\displaystyle \int_0^\infty $sin$ (x^p) dx$. Does it converge when $p<0$? Does it converge when $p>1$? My Work: Let $x^p=y$, then $\displaystyle \int_0^\infty $sin$ ...
1
vote
3answers
77 views

If $a_n\sim b_n$ and $b_n\to 0$, then $a_n\to 0$?

Let $(a_n)$ and $(b_n)$ be two sequences with $$ \lim_{n\to\infty}\frac{a_n}{b_n}=1, $$ which one usually writes as $a_n\sim b_n$. Let $b_n\to 0$ as $n\to\infty$. My question is, if then $a_n\to 0$, ...
1
vote
2answers
86 views

Convergence of $ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $

Show that the improper integral $$ \int_{-\infty}^\infty \cos(x\log(\lvert x\lvert ))\,dx $$ is convergent. I rewrote it, using even function symmetry of cosine, as twice the integral from zero to ...
2
votes
2answers
60 views

Convergence of a particular summation [duplicate]

Does summation $$\sum_{n=1}^\infty\frac{1}{n(\log n)^{a}}$$ converge if $a>1$?
3
votes
4answers
91 views

Convergence of series of $\sin(x/n^2)$

For the convergence of series of $\sin\left(\frac{x}{n^2}\right)$, is it enough to say that since, for large $n$, $$a_n:= \sin\left(\frac{x}{n^2}\right) \approx b_n:= \frac{x}{n^2},$$ so that ...
5
votes
3answers
569 views

Does $\sum_{n\ge0} \sin (\pi \sqrt{n^2+n+1}) $ converge/diverge?

How would you prove convergence/divergence of the following series? $$\sum_{n\ge0} \sin (\pi \sqrt{n^2+n+1}) $$ I'm interested in more ways of proving convergence/divergence for this series. My ...
0
votes
3answers
28 views

What is the product of $p_i-1 \over p_i$ [duplicate]

I am trying to find the value of $\prod_{i=0}^{\infty}{p_i-1 \over p_i}$ = ${\lim_{x \to \infty}} {\phi(p_x!) \over p_x!}$ Where $p_x!$ is the $x$th primorial, and $p_i$ is the $i$th prime number. I ...
4
votes
0answers
52 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \sum_{-\infty}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + ...
1
vote
1answer
55 views

Convergence of a quotient of series,

I apologize in advance for the poor formatting of this question - I'm in a coffee shop, on my phone since campus buildings closed early today.. Given: $\{a_n\}$ and $\{b_n\}$ are both positive ...
1
vote
2answers
25 views

Convergence test for Harmonic (ish) sum

I would like to test the convergence of $$\sum_{n=1}^{\infty}\frac{w^n}{n^{1/3}}$$ Where $w=\exp(2\pi i/3)$ Are there any methods that immediately come to mind?
1
vote
1answer
88 views

what value series $\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$ converges?

I have been wondering about series of $$S=\sum_{r=1}^\infty\frac{\lfloor rp \rfloor}{2^r}$$ where p is a constant positive real number and $\lfloor\cdot\rfloor$ is floor function. I know it converges ...
2
votes
1answer
65 views

Convergence of monotone boolean network in the worst case

I'm looking for (upper bound) convergence of increasing monotone boolean network (network composed only with OR, AND, identity ($f_i(x)=x_j$) functions) in asynchronous updating mode. It means that if ...
-3
votes
3answers
58 views

Convergence of a series $\sum_{n=1}^\infty x^{n-1}$ [closed]

Convergence of a series does this series converge? $$\sum_{n=1}^\infty x^{n-1}$$
1
vote
1answer
30 views

Difference among convergence almost surely, in probability, and in distribution. And advantages.

I know the definition of the three convergences. My question is, in statistical, machine learning and/or engineering applications, do the stronger convergences have any advantage over the weaker ...
2
votes
1answer
35 views

Showing series converge by comparison

Show that the following series diverges $$\sum^{\infty}_{n=1} \sin{\left(\frac{1}{n}\right)}.$$ We do this by the comparison test. Let $a_{n} = \sin{\left(\frac{1}{n}\right)}$ Now the only test I ...
0
votes
2answers
55 views

Show that two power series have the same radius of convergence

Show that the power series $\sum_{n=0}^{\infty}{c_{n}x^{n}}$ has the same radius of convergence as $\sum_{n=0}^{\infty}{c_{n+m}x^{n}}$ for any positive integer $m$. Should I use the Ratio Test? Or do ...
0
votes
0answers
34 views

rate of convergence

Good day, Please help me on this. I am not good on numerical analysis. I want to find the rate of convergence of these sequences: Let $a_1=0.9009009009$, $a_2=0.8815232722$, $a_3=0.4098360656$, ...
4
votes
1answer
55 views

Convergence of power series with eventually constant coeffcients

Assume I have a sequence $f_n$ of power series of the form $$ f_n(x) = \sum_{i=0}^\infty{a_{n,i}x^i},\quad a_{n,i}=\begin{cases}\alpha_{n,i} & n\leq i,\\b_i & n>i.\end{cases}.\tag{*} $$ ...
2
votes
1answer
92 views

Reasons for convergence

I am interested to know if anyone can see the reasoning behind the convergence $$\int_{\Omega}c(u_{k},\nabla u_{k})(u_{k}-u)dx \rightarrow 0$$ in equation (2.82), page 50 in the following book ...
3
votes
2answers
44 views

The convergence of $ \sum_{n=2}^{+\infty} \frac{1}{\sqrt{n}} \ln(\frac{n+1}{n-1})$

Good morning, I have tried to prove the convergence with the application of the criterion of comparison. I have used this increase: $$ \sum_{n=2}^{+\infty} \frac{1}{\sqrt{n}} ...
0
votes
2answers
24 views

How to use the comparison test to show that this series diverges?

I have the following series: $\displaystyle\sum_{m=2}^\infty \left( \displaystyle\frac{5}{7 \, m + 28} \right)$ The partial sums are obviously smaller than the harmonic series, but that doesn't allow ...
2
votes
1answer
32 views

Converging exponentially in probability implies convergence with probability one?

When I read the paper On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimate, the theorem 1 in it states something like If for every $\epsilon > 0$ there exists ...
2
votes
2answers
46 views

What is the definition of closed subspace?

I am trying to understand what is intended with closed subspace, I took the following guess: A closed subspace $M$ of a Hilbert space $H$ is a subspace of $H$ s.t. any sequence $\{x_n\}$ of elements ...
1
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0answers
88 views

Spectrum: Continuous?

Problem Given a Banach algebra with unit $1\in\mathcal{A}$. Consider a sequence: $A_n\to A$. Then the spectra may not converge as sets: ...
4
votes
3answers
87 views

Convergence of the series $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$

I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$ the only idea that crossed my ...
0
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0answers
39 views

Kuratowski Theorem on uniform convergence

There is a nice criterion of uniform convergence of the sequence of continuous functions due to Kuratowski: as far as I remember this involves checking whether $f_n(x_n) \to f(x)$ where $x_n \to x$ is ...
1
vote
1answer
81 views

Continuous Nowhere Differentiable Function [closed]

Define a function $\,f:\mathbb{R}\rightarrow \mathbb{R}_{+}$ by: $$ f(x)=\left|x-2\,\left \lfloor \frac{x+1}{2}\right \rfloor \right|. $$ Here are some known properties about the function $f$: ...
0
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0answers
20 views

The Convergence of Coordinate Descent involving multiple variables

Given a convex, but not differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ can be decomposed into two parts, namely, $f(x) = g(x) + \sum_{i=1}^n h_i(x_i)$, where $g$ is convex and ...
1
vote
1answer
45 views

$0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

this is related to that one the limits of $a_n $and $b_n$ Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
1
vote
2answers
45 views

the limits of $a_n $and $b_n$

this is related to that one $a_n$ is bounded and decreasing Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
2
votes
3answers
58 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 ...
1
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3answers
43 views

$b_{n}$ is increasing

I think there is misunderstanding in my last post because its contain three questions so i will post question by question step by step An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi ...
7
votes
2answers
147 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
0
votes
1answer
34 views

If $(f_n)_n$ and $(g_n)_n$ converge stochastically to $f$ and $g$, then $(f_n+g_n)_n$ converges stochastically to $f+g$

Let $(\Omega,\mathcal{A},\mu)$ be a measurable space $(E,d)$ be a separable metric space $f,g,f_n,g_n:(\Omega,\mathcal{A})\to(E,\mathcal{B}(E))$ measurable $(a_n)_{n\in\mathbb{N}}\subseteq E$ We ...
1
vote
3answers
87 views

Sequence and series problem

How do I show that the sequence $(x_n)$ defined by $$x_ {n+1} = \left(1-\frac{1}{n}\right) ^2 x_n + \frac{1}{n}, \forall \,n \in \Bbb{N}-\left\{0\right\}$$ converges? and to what limit?
1
vote
1answer
36 views

Let $\{f_k\}$ be a sequence of non-decreasing fcns. If $\int_X f_1^- d\mu <\infty$ then show $\lim_k \int_X f_k d\mu = \int_X \lim_k f_k d\mu$

I need your help to understand and analyse the following problem: Q: Let $\{f_k\}$ be a sequence of non-decreasing measurable function on $(X,\mathcal{A})$ and $\mu$ be a positive measure. If $\int_X ...
1
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2answers
43 views

If $X, X_1, X_2, \ldots $ are positive and $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$

Let $X, X_1, X_2, \ldots $ be positive random variables. Prove that if $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$ My attempt: I tried to truncate $E(|X_n-X|)$ ...
0
votes
3answers
64 views

find sum of ln(1- $\frac{1}{n^2})$. prove it converges [duplicate]

![enter image description here][1] $$ \sum = \ln(1 - \frac{1}{4} ) + \ln(1 - \frac{1}{9} ) + \ln ( 1 - \frac{1}{16}) = \ln(\frac{3}{4}) + \ln (\frac{8}{9}) + \ln ( \frac{15}{16}) = -.287 + -.117 + ...
2
votes
3answers
76 views

When does this integral converge? $\;\;\int_0^\infty {\frac{e^{-ax}}{x^2+1}}\,\mathrm{d}x$

$$\int_0^\infty \frac{e^{-ax}}{x^2+1}\,\mathrm{d}x$$ - $a$ real, for which a does this converge? (The final answer is $a\ge 0$) I've tried doing this by parts and it seems to work at first, but then ...
5
votes
1answer
79 views

Is this series $\sum_{n \geq 2}\sqrt{a_n}\frac{n^{a_n}-1}{\ln n}$ always divergent?

Let $\displaystyle \sum a_n$ be a series with positive terms which is a convergent series and suppose that we don't have $\displaystyle a_n \ln n \rightarrow 0$. Is the following series always ...
1
vote
0answers
20 views

Is this a viable generalization of Newton series?

I wonder if the following formula a viable generalization of Newton's series. $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty ...
1
vote
1answer
38 views

convergent subsequences etc.

It is well known that: A sequence of real numbers converges $\{a_n\}$ to p, if and only if every subsequence $\{a_{n_k}\}$ converges to p. I am wondering if this similar statement holds.: ...
0
votes
1answer
35 views

Question about convergence proof, why has he chosen the parameter this way

In this proof he says that $n > 2k$, but would it work if $n \ge k$, if not, why? If $p>0$ and $\alpha$ is real, then $\displaystyle\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$. Proof: ...
3
votes
5answers
78 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 ...
2
votes
1answer
85 views

Integral of the convolution of two functions: $\int_{-\infty}^{\infty} (f*g)(x)dx$

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
0
votes
1answer
59 views

What is the radius of convergence of the power series $\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$?

What is the radius of convergence of the power series? $$\sum_{n=0}^\infty \frac {(n!)^k}{(kn)!}z^n$$ Progress Used the ratio test, but got $0$ from it.
2
votes
1answer
18 views

Convergence in probability given that covariance matrix goes to $0$

Suppose I have a sequence of random vectors $\{X_n\}$ each of dimension $2\times 1$. Suppose also that I know $$ ...
0
votes
1answer
69 views

Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in ...