Convergence of sequences and different modes of convergence.

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2
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30 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
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1answer
23 views

convergence of $a_n = e^{nzi \pi /4}$

If we have series $(a_n)_{n=1}^{\infty}$ where $a_n=e^{nzi \frac{\pi}{4}}$. Where does this series convergence/divergences? If I do the ratio test I get \begin{align} r&=\lim_{n\to \infty} ...
0
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0answers
26 views

Almost sure downward convergence with some conditions implies convergence in $L^1$

If $X_n\downarrow X$ a.s., each $X_n$ is integrable and $inf_n E[X_n] > -\infty$, then $X_n \rightarrow X$ in $L^1$. As far as I know, "$X_n\downarrow X$ a.s." means that for every n, $X_n ...
2
votes
3answers
139 views

Absolute convergence tests

Hi I am interested in the following series: $$\sum_{n=1}^{\infty}(-1)^{n}(\sqrt{n+3}-\sqrt{n})$$ I have been able to show that this series converges by proving the Leibnitz test. Does anyone know ...
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3answers
35 views

Integral convergence and limit question

I have a question that has come up in a group homework project that neither I nor my partner have any idea how to solve. I'm hoping someone can give me a hint or some guidance as to how to go about ...
2
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3answers
52 views

Leibniz test for convergence of non alternating series

I am aware that one can use the comparison test and the integral test to show that the series $$\sum_{n=1}^{\infty}\frac{1}{n(n+3)}$$ converges. Is it possible to use the Leibniz test to show that the ...
3
votes
3answers
54 views

Convergence of a series with alternating denominator - Real Analysis

Decide if the series converges absolutely, conditionally, or not at all. \begin{equation} \sum_{n=1}^{\infty}\frac{(-1)^n}{(2+(-1)^n)n} \end{equation} I'm having a lot of trouble with this one. I ...
0
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1answer
23 views

Correlation between convergence radius of complex series

What do we know about the convergence between complex power series that look almost the same? For instance, if we have series $\sum_{n=1}^{\infty}a_n z^n,$ $\sum_{n=1}^{\infty}a_n z^{n+1},$ and ...
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2answers
45 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
3
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1answer
45 views

Confusion regarding almost sure events. If given infinite time, will a discrete-time gaussian process cover the entire real line?

This question really pertains to any discrete time continuous-valued, stationary stochastic process on the real line, but the Gaussian process will be adequate for this question. I have this ...
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0answers
13 views

Convergence of a monotonically increasing sequence

Given $a_n$ monotonically increasing and $a_n>0$. Which of the following converges? (1) $\frac{1}{a_n^2}$ (2) $e^{-a_n}$. I could not see any reason why both (1) and (2) will not converge. If ...
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2answers
43 views

Does $X_n \xrightarrow{\text{in distr.}} X$ and $|X_n|\leq Y$ imply $|X|\leq Y$?

We know that $$X_n \xrightarrow{\mathbb P} X \text{ and } |X_n|\leq Y \implies |X|\leq Y \text{ a.s.}$$ I was wondering if the same holds in case of convergence in distribution. So far, I've shown ...
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0answers
31 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
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1answer
23 views

Prove that the square root and exponent of a function in a $\limsup$ equals the the square root and exponent of a $\limsup$ of the function?

By what property do the following equalities hold? \begin{align*} \frac{1}{\limsup\limits_{n \to \infty} \sqrt[2n]{\left|a_n\right|}} = \left( \frac{1}{\limsup\limits_{n \to \infty} ...
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4answers
27 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
0
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1answer
52 views

Does the series $\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $ converges conditionally? [closed]

Which convergence tests can I use in order to show that the series $$\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $$ converges conditionally.
0
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3answers
50 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
2
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3answers
68 views

$\sum (\frac{1-2n}{6+2n})^n $ converges?

Verify if $$\sum_{n=0}^{\infty} \left(\frac{1-2n}{6+2n}\right)^n $$ converges The root test is inconclusive and the limit of the general term is 0. I think I should use the comparison test, in this ...
0
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1answer
36 views

Sequence of functions that converges a.e. but not in the $L^1$ norm [closed]

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
4
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5answers
128 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
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2answers
375 views

Convergence of a series 1/(2n+1)

I'm looking for a way to get an estimate on a sum of the following series: $$\sum_{i=1}^{n} \frac{1}{2i-1}$$ My exact question would be the solution for $n=500$ but I'd be interested in the generic ...
2
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1answer
34 views

The characteristic function induced by L^1 convergence function

Assume $f_n\in L^1(\Omega)$, $f\in L^1(\Omega)$, and $f_n\to f$ in $L^1(\Omega)$ where $\Omega\subset R^N$ is open. Define $$ E_t^n:=\{x\in\Omega, f_n(x)>t\}.$$ Hence we have \begin{equation} ...
0
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1answer
15 views

How to use the triangle inequality to get $d(x_k,y)\ge d(x,y)-d(x_k,x)$ (Prove limit of a sequence is unique)

Here, $d$ is the Euclidean distance: And the triangle inequality in terms of $d$ is: I have no idea how to come up with the $d(x_k,y)\ge d(x,y)-d(x_k,x)$ inequality. What I've tried: ...
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2answers
25 views

Convergence/divergence test for infinite integrals

What would be a suitable test for convergence or divergence of the series: $$\sum_{k=1}^{\infty} \frac{k}{k^3+1}?$$
4
votes
2answers
80 views

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0.

Prove that the sequence $\left\{\frac{n}{n^2+1}\right\}_1^\infty$ converges to 0. I'm missing something. I doubt that I'm really doing anything from " $\frac{1}{n+1}$ " on. I set it $< ...
2
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1answer
32 views

Convergence of alternating series not subject to alternating series test

Given two $a_n,\; b_n>0$ such that $lim_{n \to \infty} a_n,\; b_n=0$ and $\lim_{n \to \infty} \frac{a_n}{b_n}=1$, where neither series is necessarily monotonic: if $\displaystyle \sum_{k=1}^\infty ...
2
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5answers
109 views

Determining convergence or divergence of series

I am wondering the convergence or divergence following series $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{ n+\sin (n)} \\ $$ My 1st attempt is 'alternating series test' $$ $$ But, ...
2
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1answer
33 views

Proving that $\int_0^t s(x) dx = \frac{t^2}{1-t}$

Let $ s =s(x)$ given by $s(x) = \sum_{k=1}^\infty (k+1)x^k. $ Prove that for all $ t \in ]-1,1[,$ $$\int_0^t s(x) dx = \frac{t^2}{1-t}$$ Conclude that, for all $x \in ]-1,1[,$ $$\sum_{k=1}^\infty ...
3
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1answer
38 views

Does 'bounded by convergent series' imply convergence?

Suppose we have a real series $\sum x_{n}$ which is convergent. If we have either $0<y_{n}<x_{n}$ or $x_{n}<y_{n}<0$ for all $n$ past some limit (so $|y_{n}|<|x_{n}|$, and they have ...
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1answer
31 views

Determining Rate of Convergence

I have a question from the homework here: Show that the following sequence converges linearly to 0 $$P_n = \frac{1}{n^2}; n \ge 1$$ So we know $$\lim\limits_{x \to \inf} \frac{|p_{n+1} - ...
2
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2answers
54 views

A convergence test similar to Gauss' test.

Consider a sequence of complex numbers $(a_n)$ and assume that we can write $$\frac{a_{n+1}}{a_n}=1+\frac{\lambda}n+\frac{b_n}{n^2}$$ where $b_n$ is bounded and $\Re(\lambda)<-1$. Can we show that ...
0
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1answer
53 views

Convergence of $\sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2}$

Does the series $$ \sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2} $$ converge? The ratio test is inconclusive, so I think I must use the comparison test. But I couldn't ...
7
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0answers
66 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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2answers
58 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
4
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1answer
107 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
2
votes
2answers
28 views

Suppose that $f(x) \ge 0$ and $\lim_{x \to c} f(x) = L$. Prove $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$

Suppose that $f(x) \ge 0$ in some deleted neighborhood of $c$, and that $\lim_{x \to c} f(x) = L$. Prove that $\lim_{x \to c} \sqrt{f(x)} = \sqrt{L}$ under the two different assumptions on $L$: $L=0$ ...
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1answer
15 views

Convergence of an Iterative Sequence…

Let $g(x)=\frac{2+x}{1+x}$. Set now the sequence $(x_n)_{n\in\Bbb{N}}$ such that $x_0=0$ and $x_{n+1}=g(x_n)$. Show that this sequence converges and, furthermore, converges to $\sqrt[]{2}$.
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0answers
18 views

Convergence of a Double Sum over 2 integers

Does the following double summation over x, x' (both integers) converge? $\sum\limits_{x=-\infty}^\infty \sum\limits_{x'=-\infty}^\infty \frac{Sin^2(2 \pi(x-x'))}{(x-x')^2}$. If so evaluate the sum. ...
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1answer
29 views

Cauchy sum of ratio of sequences not converging.

Prove if ${x_n}$ and ${y_n}$ are Cauchy and $x_n + y_n > 0$, for all natural n, then $\frac{1}{x_n + y_n}$ cannot converge to zero. Attempt: Suppose $x_n → a$ and $y_n → b$. Then $x_n + y_n → a + ...
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1answer
18 views

Bounded intervals, sequentially compact.

Can someone please give me an example of a bounded interval in R that are not sequentially compact? A subset E of R is said to be sequentially compact if and only if every sequence x_n in E has a ...
3
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1answer
53 views

Proving convergence of $\sum \frac{\sin n}{2^n}$

Prove that the following sequence ($x_n$) is convergent: $$ x_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} + ... + \frac{\sin n}{2^n} $$ I have tried to use to the sequence is ...
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1answer
20 views

Radius of Convergence (Non-Series)

I am confronted with the following exercise: Compute the radius of convergence for the expansion at the point $z=4+4i$ for \begin{equation} f(z)=\frac{z^{5}e^{z}}{(2-z)(3i-z)} \end{equation} I ...
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0answers
10 views

Convergence of normalized stochastic integral

I am wondering about some results about the convergence of processes like that : $$ \frac{1}{T} \int_{0}^{T} H_{s}dM_{s} $$ with $M_{s}$ a semi-martingale when T goes to $ +\infty$ Thanks a lot :-) ...
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1answer
27 views

Converse of alternating series test

Is the converse of the alternating series test true? In other words, given a sequence $a_n>0$, with neither $a_{2n}$ nor $a_{2n-1}$ constant, for which there exists no positive $N$ such that ...
0
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1answer
28 views

Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
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0answers
24 views

Convergence of probability density in the tail.

Let $f(x)$ be a probability density with respect to the Lebesgue measure. The distribution has first moment, e.g. $\int_{-\infty}^\infty |x| f(x) dx < \infty$. Further assume that there exists $K$ ...
0
votes
2answers
37 views

Convergence of square of monotone sequence implies convergence of sequence.

Suppose $(a_n)$ is a monotone sequence. Prove that if $(a_n^2)$ is convergent, then so is $(a_n)$. How do I use the monotone convergence theorem to prove this?
6
votes
2answers
132 views

How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?

I'm trying to show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m)) + \cdots$ converges for all real numbers $m$. To be specific, the series is defined as follows: $\sum_1^\infty{a_k}$ ...
1
vote
2answers
26 views

$f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
0
votes
1answer
29 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...