# Tagged Questions

Convergence of sequences and different modes of convergence.

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### convergence of power series, expanded by maclaurin

I'm getting ready for a test and I stumbled upon a question which goes like this: A function f(x) is given and I need to expand it to a power series using Mcloren sequences and then calculate its area ...
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### What is the rate of convergence of Brownian motions Increments?

Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT?
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### closeness of matrices

I'm really lost in math and would really appreciate any help with the following problem. Denote as $S_{+}(p)$ the set of all positively defined symmetric real-valued matrices of size $p \times p$. ...
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### When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
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### Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...
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### Convergence of sequence $a_0 := b$ and $a_{n+1} = 2^{a_n}$ in $\hat{\mathbb{Z}}$.

I have a question about the convergence properties of a sequence in $\hat{\mathbb{Z}}$, the completion of $\mathbb{Z}$. It is part of an exercise is due to this syllabus. I got confused somewhere. ...
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### On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
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### On a first set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of this claims and show where were my mistakes or inaccurancies? Also ...
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### Convergence test of $\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$

I know, there are some threads dealing with this sum but I want to solve it with the integral test for convergence(more) $$\sum\limits_{n=3}^{\infty} \frac{1}{n\log(n)\log(\log(n))}$$ I can't ...
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### Can you prove the convergence of $\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx$?

Can you prove the following improper integral is convergent? $$\int_0^{1/2}\frac{\sin x}{x}\frac{1}{\log\frac{1}{x}}\bigg(1+\frac{1}{N}\log\frac{1}{x}\bigg)^N\,dx.$$
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### examine the convergence of the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{n+(-1)^{n+1}}$

I have no idea how to examine the convergence of the series: $$\sum_{n=0}^{\infty} \frac{(-1)^n}{n+(-1)^{n+1}}$$ We can see that $\frac{(-1)^n}{n+(-1)^{n+1}} \to 0$. However we can't use criterium ...
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### convergence of improper Integral..

I need help finding if the improper integral below converges. $$\int _{ 2 }^{ \infty }{ \frac { dx }{ \sqrt [ 3 ]{ 1-{ x }^{ 4 } } } }$$. we learnt at class: comparison test ratio test Thanks ...
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### Convergence of given sequence

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function with: $$f(x) = x - \arctan{x}$$ We consider the sequence $(x_{n})$ with $x_{0} > 0$ and $x_{n + 1} = f(x_{n})$, for any $n \in \mathbb{N}$...
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### On calculations for $\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right)$, where $\mu(n)$ is the Möbius function

I define for some set of real numbers $x\in S$ (see that it is my Question 1.) the domain of the function $$f(x)=\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right) ,$$ where $\mu(k)$ is the ...
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### Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert}$

I have this exercise but not sure if I'm doing it right $$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert}$$ I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert}$ then ...
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### Convergence of the series $\sum_{n=1}^{\infty}(-1)^n\ \frac{n\log(n)}{e^n}$

Is the series $\sum_{n=1}^{\infty}(-1)^n\ \dfrac{n\log(n)}{e^n}$ convergent or divergent? How can I solve this question? Please Help. Thank you.
### What is the interval of convergence of : $\sum_{n=1}^\infty\frac{n^n}{n!}x^n$?
$x+ \frac{2^2x^2}{2!}+ \frac{3^3x^3}{3!}+ \frac{4^4x^4}{4!}+...$ Possible answers- 1.($0,1/e$) 2.(1/e, $\infty$) 3.(2/e, 3/e) 4.(3/e, 4/e) ...