Tagged Questions

Convergence of sequences and different modes of convergence.

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Central limit theorem in multidimension with unknown covariance

Let $X_1,\dots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment. If $d=1$, $\bar{X}_n=1/n\sum_{i=1}^nX_i$ and $S_n=1/(n−1)\sum_{i=1}^n(X_i−\bar{X}_n)^2$ then ...
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Show that if $\sum_{k=1}^m c_k =0$, $\sum_{n=0}^{\infty} \sum_{k=1}^m \frac{c_k}{nm+k}$ converges.

This is a generalization of this: Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Here is my solution. To show that if $\sum_{k=1}^m c_k =0$, ...
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Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
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Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$ Note : In wolfram alpha show this result and in the same time by ratio test ...
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Determine if this series $\sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges

Determine if the following series converges: $$\sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}.$$ (http://i.stack.imgur.com/qWiuy.png) I don't know how to start.
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Determine if $\sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}$ congerges

Determine if the following series converges: $$\sum_{n \geq 2} \frac{1}{\sqrt[n]{\ln n}}.$$ I'm supposed to use here the limit comparison test, but I don't know how to choose the second series.
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Is it always possible to converge from an integer to another integer? [closed]

Let's say I'm given a fixed integer, I. I'd like to know if it is always possible to find a function, that starting from any random integer J will converge to or oscillate reasonably close (let's say ...
140 views

Why doesn't this work for Rudin Exercise 3.8

The problem is 3.8 exercise in baby Rudin: If $\sum{a_n}$ converges and $\{b_n\}$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. Why can't I just do this?: Let $M$ be an ...
Integral Convergence $\sin{x}/x^{3/2}$
Does the following integral converge: $$\int_0^\infty{\frac{\sin x}{x^{3/2}}}dx$$ I have tried to integrate this by parts and arrived at: $$-x^{-3/2}\cos x -\int \frac 12{x^{-1/2}}\cos{x} dx$$ ...