Convergence of sequences and different modes of convergence.

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9
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2answers
260 views

Convergence of $\sum _{k=1}^\infty \sin \left(\sqrt{k}\right)/k$

This was asked at an oral examination. Does the series $\displaystyle \sum _{k\geq1}\frac{\sin\left(\sqrt{k}\right)}{k}$ converge ? After playing with Mathematica, it's very likely it ...
0
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4answers
48 views

Does this sequence converge

Does the sequence $x_n=3^n−2^n$ converge? I can show that it is increasing but how to show that it is bounded?
1
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1answer
34 views

If $f_n$ converges uniformly then $\cos(t)f_n$ converges uniformly?

In my Fourier Series course, it seems the following result is used: If a series of function $\sum a_n(t)$ converges uniformly then the sequence of functions $\cos(t)\sum a_n(t)$ converges ...
0
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1answer
35 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
0
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1answer
31 views

Question on regulated functions

Suppose that $f:[a,b]→\mathbb{R}$ is a regulated function. Then the integral $\int_a^bf(x)dx$ is defined as the limit $\lim_{n→∞}\int_a^bs_ndx$, where $(s_n)_{n∈\mathbb{N}}$ is a sequence of step ...
0
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2answers
67 views

Prove $f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ defines a continuous function on $\mathbb{R}$.

Prove $$f(x)=\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k$$ defines a continuous function on $\mathbb{R}$. I think we can show that if $\sum_{k=1}^\infty \frac{1}{k!\sqrt{k}}x^k $ is uniformly ...
0
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1answer
39 views

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

Does $\sum_0^\infty(\frac{1}{9n+1})$ converges? If yes, then to what?
0
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0answers
14 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 : ...
1
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3answers
53 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 ...
2
votes
2answers
139 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 $f$ ...
1
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2answers
46 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 ...
1
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1answer
39 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 ...
1
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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.
3
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0answers
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 $z\in A?$ ...
1
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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 $u_{n_j}$ ...
1
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1answer
25 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, ...
-1
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0answers
32 views

An example for such spaces and an example for infinite-dimensional Hilbert spaces

It is known that in any finite-dimensional Hilbert space, the weak topology and the strong topology coincide; I need an example for such spaces and an example for infinite-dimensional Hilbert spaces ...
1
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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 ...
1
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1answer
16 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 $x\in D$, ...
1
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1answer
23 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 ...
1
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1answer
66 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?
0
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0answers
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 ...
1
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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 ...
0
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1answer
80 views

Topology Hausdorff $\Leftrightarrow$ Unique Limits [closed]

Prove that a topology is Hausdorff iff every net (not filter!) has at most one limit.
0
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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?
-1
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3answers
63 views

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 ...
3
votes
1answer
52 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: ...
0
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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.
1
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0answers
28 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
votes
1answer
176 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. ...
0
votes
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): ...
0
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0answers
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. $$\lim_{n\rightarrow \infty }\prod_{i=1}^{n}\frac{1}{ab^{i-1}+1}$$ $a>0$, $0<b<1$, ...
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 ...
0
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1answer
28 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 ...
0
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1answer
51 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 ...
1
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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 $\displaystyle\int f(y)^{-1}dy=yf(y)^{-1}-F(f(y)^{-1})+C$ ...
0
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1answer
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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0answers
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 ...
0
votes
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}) ...
0
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0answers
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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0answers
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, ...
19
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1answer
247 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 $0<a_n<1$, $n=1, 2, 3\dots$ ?
1
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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 ...
1
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1answer
38 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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0answers
54 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 ...
1
vote
1answer
22 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 ...
0
votes
0answers
21 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 ...