Convergence of sequences and different modes of convergence.

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Does $\sum_0^\infty(\frac{1}{9n+1})$ converges?

Does $\sum_0^\infty(\frac{1}{9n+1})$ converges? If yes, then to what?
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15 views

How to determinate the convergence interval D'Alembert

I need to determinate the interval of convergence using the generalized criterion of D'Alembert. Considering $\sum\limits_{n=1}^\infty nx^{n-1}$ To determinate the interval of convergence, I did : ...
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3answers
54 views

Convergence of real sequence ${\frac{x_n}{n}}$ as $n$ tends to $\infty$ [duplicate]

Let ${x_n}$ be a real sequence such that $\lim_{n\to\infty}(x_{n+1}-x_n)=c$. Then, talk about convergence of the sequence ${\frac{x_n}{n}}$ My try: I did not understand how to proceed. I thought of ...
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2answers
140 views

Can a function be square integrable without being integrable?

Reading Tolstov's 'Fourier Series', which states that $f(x)$ is square integrable if both $f$ and its square both have finite integrals over some interval. I haven't seen this restriction on $f$ ...
2
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2answers
52 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
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1answer
40 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
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1answer
64 views

Check the uniform convergence of parametric integral

While $0< \alpha <+ \infty$, prove if the parametric integral is uniform convergent on $\alpha$'s domain: $$\int^{+ \infty}_{0} e^{- \alpha x} \sin \beta x dx$$ $\beta$ is nonzero constant.
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69 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
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1answer
26 views

Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
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1answer
28 views

Types of Convergence (Random Variables)

Suppose that for every $n\ge 1$, the law of $X_n$ is given by $P[X_n=n^2]=\beta_n$ and $P[X_n=0]=1-\beta_n$, determine if $(X_n)_{n\ge 1}$ converges in probability, in $L^1$ or almost sure to zero, ...
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3answers
27 views

What can we say about the convergence of this series

$$\sum {z^n\over n!} $$ I used Alembert's Ration test and get $$\lim_{n \infty}{u_n\over u_{n+1}}={n+1\over z}$$ As this tends to $\infty>1$ can i say that the given series is convergent for all ...
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1answer
18 views

convergence in measure implies the composition of the sequence of functions and a continuous function also converges in measure

Let $D$ be a measureble set in $\mathbb{R}^n$. Suppose $\mu(D)<\infty$. Let $\phi: D\times \mathbb{R}\to \mathbb{R}$ be a continuous function such that for almost every $x\in D$, ...
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1answer
24 views

Borell Cantelli Application

If i got that $\mathbb{P}(\underbrace{|X_{n}|>n^{\frac{1}{2}+\epsilon}}_{=:A_{n}})\leq \exp\left(-\frac{n^{2\epsilon}}{8}\right)$ with $\epsilon \in (0, 0.5)$. I know that ...
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1answer
67 views

Does this converge? $\sum_{n=3}^\infty \frac1{n+ \log(n)}$

This seems too easy, my friend said he couldn't get it. maybe I am wrong?? $$\sum \limits_{n=3}^\infty \frac1{n+ \log(n)} \leq \frac1{n+n}=\frac1{2n} \leq \frac1n$$ Which converges as harmonic?
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50 views

My first integral test, is it correct?

I want to test for convergence on $\sum \limits_{n=3}^\infty \cfrac1{n(\log n)(\log(\log n))}$ Now I have just learnt the integral test off of a fellow stack exchange user(M. Vinay). $\int_3^\infty ...
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2answers
34 views

I want to prove a series converges absolutely

I want to show that: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}}$ converges absolutely. Now, here is what I have done: $\sum \limits_{n=1}^\infty \cfrac{n^2 + \cos(n)}{e^{n^3}} \leq ...
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1answer
81 views

Topology Hausdorff $\Leftrightarrow$ Unique Limits [closed]

Prove that a topology is Hausdorff iff every net (not filter!) has at most one limit.
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1answer
28 views

arithmetic of converge series question [duplicate]

If $\sum _{n=1}^{\infty \:}a_{n\:}$ converges and $\sum _{n=1}^{\infty \:}b_{n\:}$ converges. how to proof that $\sum _{n=1}^{\infty \:}a_{n\:}-b_n$ also converges?
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Convergence of $\sum_{i=1}^\infty \sin^2(\frac{1}{i})$ and $\sum_{i=1}^\infty\cos^2(\frac{1}{i}).$

I need to check convergence of these sums: $$\sum\limits_{i=1}^\infty \sin^2\left(\frac{1}{i}\right)\qquad\sum\limits_{i=1}^\infty\cos^2\left(\frac{1}{i}\right).$$ Does comparing these sums to ...
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1answer
53 views

What is the proper way to handle the limit with little-$o$?

I was hoping to show that $$\left(1-\frac{x}{n}+o\left(\frac{2x}{n}\right)\right)^n \xrightarrow{n\to\infty} e^{-x}$$ which would be just fine without the little-$o$. Trying binomial formula: ...
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4answers
65 views

Testing the convergence of a series [closed]

Does the following series converge? $$\sum_{n=0}^{\infty}\frac{1}{n+3}$$ Please explain with any convergence test you used.
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32 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
3
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1answer
186 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
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2answers
34 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
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36 views

Evaluating $\lim_{n\rightarrow \infty }\prod_{i=1}^{n}\frac{1}{ab^{i-1}+1}$

I'm sure that this function must converge to a constant but I can't write it in a closed form. $$\lim_{n\rightarrow \infty }\prod_{i=1}^{n}\frac{1}{ab^{i-1}+1}$$ $a>0$, $0<b<1$, ...
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1answer
23 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
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29 views

Show that $\mu(f_n^+) \rightarrow \mu(f^+) $ and $\mu(f_n^-) \rightarrow \mu(f^-) $, using Fatou's Lemma.

I'm starting learning about Fatou's lemma. How would you apply it to solve the following problem: Let $g^+ = max (g,0)$ and $g^- = max (-g,0)$. Let $f_n$ be integrable on measure space with measure ...
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1answer
54 views

Proof for multivariate Newton-Raphson method

How can the proof for Newton's method for a single variable be extended to the multivariate version? Forgive me if this is trivial, but I don't seem to get it. Any links or proofs would be greatly ...
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1answer
57 views

How to analyze convergence or divergence of the integral $\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$

Analyze convergence or divergence of the integral $\displaystyle\int_1^{\infty}(t^2+\ln^2t)^{-1}dt$ since $\displaystyle\int f(y)^{-1}dy=yf(y)^{-1}-F(f(y)^{-1})+C$ ...
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30 views

tightness of sequence of degenerate probabilities

If $\delta_x$ denotes for $x\in \mathscr{R} $, the degenerate distribution at $x$, prove that the sequence $\delta_{x_n}$ of probabilities on $(\mathscr{R,B})$ is tight iff $x_n$ is bounded. This is ...
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33 views

Can someone please check if I have solved this convergence question correctly?

I got this problem today. Can someone please check my proof and confirm if it is correct or point out the place where it is wrong? I think it is correct, but it is so hard to see what is going on, so ...
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1answer
23 views

Existence of expected value with complex power

Suppose that $X$ be a random variable taking values on $(0,+\infty)$ with density function $f(x)$ and we have $\mathbb{E}(X^2)<\infty$. Can we conclude that $$\mathbb{E}(X^{-2-it}) ...
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14 views

Analytic result for this series

I'm having troubles in assessing whether this series in two unknowns does converge to a more tractable expression. The series is: $$\sum_{j,k}^\infty j \cdot k \cdot \phi_j \cdot \mu_{j,k} = c $$ ...
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40 views

How to show a sequence of functions does not converge uniformly

Let $f$ be a continuous function on $[0, \infty)$ such that $0\leq f \leq Cx^{-1-\rho}$, where $C$ and $\rho$ are positive constants. Let $f_k(x)=kf(kx)$. $\textbf{Question}$: Show that $f_k$ does ...
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1answer
25 views

Simpson's composite rule rate of convergence.

Hello I have wriiten a program in Matlab that determines an Integral using Simpsons rule and it also determines the rate of convergence. I tried my program on the following examples: $f(x)=\sin{ x}$ ...
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1answer
51 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
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250 views

Does the series $\sum\frac{a_n}{a_{n+1}}\frac{1}{n}$ always diverge if $0<a_n<1$?

Does the series $$\sum_{n=1}^{\infty}\frac{a_n}{a_{n+1}}\frac{1}{n}$$ diverge for any $a_n$, satisfying $0<a_n<1$, $n=1, 2, 3\dots$ ?
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1answer
41 views

Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
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1answer
22 views

Proving convergence/divergence for $p$-series

I have an exam in Calc 2 coming up. As such, I am doing previous exams given by our current professor. However, the exams lack a solution set, so I will post the question, and the answer I wrote down ...
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1answer
39 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
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55 views

Does $ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right )\right) $ converge? [duplicate]

I am trying to determine whether the $$ \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\left ( \sum_{m=1}^{n}\frac{1}{m} \right ) \right) $$ converges or not. I have tried the popular tests, but all ...
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1answer
23 views

Convergence of the sequence of operators on a Banach space.

Let $T$ be a bounded linear operator on a Banach space $E$, and let $(T_n)$ be a sequence of bounded linear operators on $E$ such that $$\| (T - T_n)x \| \to 0 \ \ \text{ as } n \to \infty$$ for all ...
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22 views

weak convergence of a sequence of cdfs

Suppose $F_n \to F $ weakly, $ x \in c(F)$ and $ x_n $ is a real sequence converging to x. Prove that $F_n(x_n) \to F(x) $. Here $F_n$, $F$ are cdfs and $c(F)$-set of continuity points of $F4. I ...
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76 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
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Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
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31 views

problem on almost sure convergence

Let {$X_i$} be iid with finite second moment. Let $Y_n = \frac {2} {n(n+1)} \sum_{i=0}^n i*X_i $, n$\ge$1 Show that $Y_n \to E(X_1) $ I tried to define $Z_i = \frac {2} {(n+1)} i*X_i $ Then $Y_n = ...
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1answer
34 views

This is a ratio test question.

How do you use the ratio test to show whether converges or diverges? Using symbolab gave me something with "series root test", however it is not covered in my course. Would it be possible to ...
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20 views

real sequence and convergence in probability

$X_n$ is a sequence of random variables.$X_n \equiv a_n$, $a_n $ is a real sequence. Then prove that $X_n $ converges in probability iff $a_n$ converges and then $X_n \to \lim_{n\to\infty} a_n$ in ...
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1answer
40 views

Alternating Series Proof

I should be able to figure this out, but it has me a bit confused conceptually. I'm really just not sure how to approach it in a rigorous fashion. Any help? If $a_0, a_1, a_2, . . .$ is a decreasing ...
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1answer
41 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...