Tagged Questions

Convergence of sequences and different modes of convergence.

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Right-const function and pointwise/uniform convergence

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. ...
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$L^p$ convergence of smooth compactly supported functions

I already checked the similar question here, but want to check slightly different argument. Given $f(x)\in L^p(\mathbb{R}^n)$, can I find $f_n(x) \in C^\infty_0$ s.t. $f_n \rightarrow f$ almost ...
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What should be the value of $\alpha$ for which the series is convergent?

The series $$\sum \frac{\log(1+\frac{1}{n})}{n^\alpha}$$ a. Converges if $\alpha>0$ b. Diverges for all $\alpha\in \mathbb{R}$ c. Converges if $\alpha=0$ d. Converges if $\alpha<0$ ...
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(Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
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Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$$
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Converging savings

Let's imagine abstract situation that there is a city where all families have houses (their number is = $1000$) which are located on the edge of a big circle so every house has exactly two neighbors ...
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Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
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What condition on the coefficients $a_n$ will guarantee $f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ is k times differentiable?

$f(x) = \Sigma_{n = -\infty}^{\infty}a_{n}e^{2\pi inx}$ What condition on the coefficients $a_n$ will guarantee $f$ is $k$ times differentiable? I'm not sure where to begin with this, because it ...
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Determine whether the following integral is convergent or divergent $\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$

Please, help me to determine the following integral: $$\int^{2}_{0} \frac{1}{\sqrt[3]{x} (x+\sqrt{x})}dx$$ As we know, this is the II order indeterminant integral. I've tried to use comparison test, ...
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Why does $\sum_{n=2}^\infty \frac{1}{\ln(n!)}$ diverge?

$$\sum_{n=2}^\infty \frac{1}{\ln(n!)}$$ I tried by comparing it to $\sum_{n=1}^\infty \frac{1}{n}$ but i seem to fail. I think I need to compare with series that are smaller and diverge. Help.
Suppose $F_n$ are a sequence of distribution functions with the property that for any measurable $g$ , we neccesarily have that \int_{\mathbb{R}} g \; dF_n \xrightarrow{n \rightarrow \infty} \int_{\... 1answer 52 views Can every element in the arbitrary space be converged to? When I have for example \mathbb R then I'm able to create a sequence which will converge to any of the elements in \mathbb R: \begin{align} \frac{1}{n} &\rightarrow 0\\ \frac{1}{n} + 1 &\... 0answers 18 views Weak Law of Large Numbers, biased expectation? I want to show that:\hat{\sigma^2}=(1/n)\sum^{n}_{i=1} ( X_i-\bar{X} )^2$$is a consistent estimator of \sigma^2. I was using the Weak Law of Large Numbers in the sense that:$$E(X_i-\bar{X })...
Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$Y(k)=\frac{1}{k}\sum_{i=1}^k X_i$$ Notice that $Y(k)$ is ...