# Tagged Questions

Convergence of sequences and different modes of convergence.

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### For which $\alpha$, $\beta$ does $\int\limits_1^{\infty} x^{\alpha} \cdot (\ln x)^\beta dx$ converge? [duplicate]

For which $\alpha$ and $\beta$ does the following integral converge ?: $$\int_{1}^{\infty}x^{\alpha}\,\ln^{\beta}\left(x\right)\,\mathrm{d}x$$ Here is my analysis: I noticed that the function ...
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### Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
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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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### Convergence sequence of mean implies convergence in mean / weakly consistence of subsequence of regression function estimates

Let $(X_n)$ be a sequence of positive random variables. Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=0$. This imply that $(X_n)$ converges to zero ...
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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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### Condition on $(a_n)_{n \in \mathbf{N}}$ for Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ to converge for $Re(s) > 1$.

I'm curious about the conditions on a sequence $(a_n)_{n \in \mathbf{N}}$ of real numbers such that the Dirichlet-series $\sum_{n = 1}^\infty \frac{a_n}{n^s}$ converges absolutely for $Re(s) > 1$. ...
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### Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?

Does the following series converge or diverge? $$\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$$ The methods I have at my disposal are geometric and harmonic series, comparison test, limit ...
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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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### Convergence of random variable 5

If $Q < \frac{n}{m^2} X_n$ where $X_n$ is a sequence of random variables, $X_n \xrightarrow{a.s}1$, $0\leq Q \leq1$, $m=\omega(\sqrt{n})$ (The $\omega$ denotes the order, see here). Then, how can ...