# Tagged Questions

Convergence of sequences and different modes of convergence.

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### Is there a nicer way to show that the series is convergent?

I'd like to show that for a fixed $z\in\mathbb C\setminus\mathbb Z$ the series $$\sum_{n=1}^\infty \left| \frac{1}{z-n} + \frac{1}{n} \right|$$ is convergent. I think, one can do it as follows. Fix ...
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### convergence in probability of division and their expected values

Let $\frac{X_n}{Y_n} \rightarrow 1$ in probability. Then does $\frac{\mathbb{E}[X_n]}{\mathbb{E}[Y_n]} \rightarrow 1$? If not, what are the conditions required for this to hold?
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### Prove the convergence of sequence

Let $f : \mathbb{R} \rightarrow \mathbb{R}$, with $f(x) = \sqrt[3]{3x^2 - 2x^3}$ Let $x_{0} \in (0, 1)$ and $x_{n + 1} = f(x_{n}), \forall x \in \mathbb{N}$ Prove that $(x_{n})$ is ...
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### Almost sure convergence implies convegence in distribution - proof using monotone convergence

I'm trying to understand the following proof of the statement : "Almost sure convergence implies convegence in distribution" The definition of convergence in distribution is given as follows : $X_n$...
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### Convergence of Ritz polynomial in mean square

If i use a method of weighted-residual or Ritz method and obtain a numerical approximation as a polynomial ... How can i prove the convergence of this solution to the exact(in mean square sense)?
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### If $a_n\ge0$ and $\sum a_n$ converges then $\sum\sqrt{a_na_{n-1}}$ converges, what about the converse?

Suppose the series $\sum_{n=1}^{\infty}{a_n}$ is convergent ($a_n \geq0$), Is it true that $\sum_{n=1}^{\infty}\sqrt{a_na_{n-1}}$ is convergent ? Is the converse true? My attempt: The first part I ...
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### Convergence of a series of bounded linear operators to zero.

I'm working on a proof and I have a term I can't seem to handle. The problem at hand can be isolated to the following: Let $(T_n)$ be a sequence of bounded linear operators that are uniformly ...
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### Evaluate convergence radius for $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$

Follow-up question (see "Edit") Given $f(x) :=$ $\sum_{n=0}^{\infty}$ $(3x - 2)^n \over 5^n(n+2)\sqrt{n+3}$, I have to evaluate the largest open interval where $f(x)$ converges. ...
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### Why does $\sum_{n=1}^\infty \frac{\cos x^n}{n}$ converge on $-1\le x \lt 1$, and uniformly converge on $(-1,0)$

Given the series: $$\sum_{n=1}^\infty \frac{\cos x^n}{n}$$ This is similar to $$\sum_{n=1}^{\infty}\frac{x^n}{n}$$ This series converges on $-1 \le x \lt 1$ and converges uniformly on $(-1,0)$ ...
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### How to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...
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### Convergence of measures in total variation sense

Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures. Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ ...
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### We can say that any Cauchy sequence converges to some value in some space?

Maybe this is a trivial question but I need to ask and clarify myself: I know that any Cauchy sequence converges to some value inside some space if this space is complete. But, we can say that any ...
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### Convergence of $\sum_{n\in\mathbb{N}_{>0}} f(n)\mathrm{log}(n)$

A function $f(n)$ has the following conditions: $$f(n),n\in\mathbb{N}_{>0}$$ $$f(n)\in[0,1]$$ $$\sum_{n\in\mathbb{N}_{>0}} f(n)=1$$ Does the following sum always converge? Or does a ...
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### a.s. convergence and conditional expectation

I have a stochastic process $X(t,\omega)$ which is a martingale. It is showed that there exists a r.v. $X(\infty)$ such that in $L^1(\Omega)$, $\lim_{t \rightarrow \infty}X(t) = X(\infty)$. In my ...
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### Convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$

Problem: Analyze the convergence of an improper integral $I=\int_{0}^\infty x^a \ln{(1+\frac{1}{x^2})} dx$. It seems to me that 'I' converges for $0<a<1$. My work: I wrote integral 'I' as a ...
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### $((n-1)^{0.5})/(((n+1)^2)-1)$ Is the sum convergent?, why or why not? [closed]

$$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks.
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### Bounding $L_p$ norms on a convergent $L_1$ sequence
I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...