Convergence of sequences and different modes of convergence.

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1answer
48 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
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2answers
44 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
1
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1answer
45 views

Is there a contradiction in these two definitions of limit Superior?

Definition 1 : Let $A=\{a_n\}$ be a sequence of real numbers not necessarily bounded . Then we define : $\lim \sup ~ a_n = \inf ~\{\sup ~a_n,~\sup ~a_{n+1}~\sup ~a_{n+2}, \cdots\} $ Definition 2 : ...
3
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1answer
45 views

Prove $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?

How to prove $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$? I would appreciate any help.
4
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1answer
102 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
0
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1answer
42 views

alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?

An alternating series $\sum\limits_{n=1}^\infty (-1)^na_n$ is divergent , $a_n\geq0$, and $\lim\limits_{n\to\infty}a_n=0$. Could we conclude that $\sum\limits_{k=1}^\infty A_k$ is divergent, too ? ...
2
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2answers
79 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
3
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1answer
59 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
2
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1answer
95 views

convergence of $ x^{x^{x^{x^{x^{.^{.^{.^{}}}}}}}}$?

From the link here, the following question was asked: What is $y'$ if $y = x^{x^{x^{x^{x^{.^{.^{.^{}}}}}}}}$? The function goods interesting. Here is my questions: What is the domain and ...
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0answers
9 views

Convergence results for incremental generalized gradient methods

Are there convergence results of incremental and stochastic subgradient / generalized gradient methods for locally Lipschitz functions that are not necessarily convex? I am mainly interested in ...
0
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1answer
37 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
3
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2answers
35 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a measure space. It is well known that a measure is continuous from below as well as from above: ...
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3answers
46 views

Convergence of a series with general term $u_n=\int_0^{\infty}e^{-x^n}dx$

I would like to find if the series $\displaystyle \sum_{n=1}^{\infty}u_n$ is convergent or divergent where $$u_n=\int_0^{\infty}e^{-x^n}dx. $$ I've tried to obtain $v_n$ with explicit form such that ...
2
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1answer
48 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
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1answer
30 views

Decreasing sequence, bounded below.

Suppose that $0 \leq x_1 < 1$ and $x_{n+1} = 1 - \sqrt{1 - x_n}$ for all natural $n$. Prove that $x_n$ is decreasing and bounded below as $n$ converges. Attempt: Suppose that $0 \leq x_1 < 1$ ...
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1answer
33 views

Does the sequence converge or diverge?

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) ...
3
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0answers
74 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
0
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0answers
60 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...
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26 views

homotopy group of the limit space

Let $V_k(\mathbb{R}^{n+k})$ be Stiefel manifold. Using $\pi_i(V_k(\mathbb{R}^{n+k}))=0$ for all $i\leq n-1$, how to obtain $\pi_i( V_k(\mathbb{R}^{\infty}))=0$ for all $i\in \mathbb{N}$? Can I just ...
0
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3answers
53 views

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then $\lim \inf (x_n) \leq \lim \inf(y_n)$

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then show that $\lim \inf (x_n) \leq \lim \inf(y_n)$ and $\lim \sup (x_n) \leq \lim \sup (y_n)$ ...
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24 views

If in an upper bounded real sequence $X : v_m =\sup \{x_n : n \geq m\}$, then $x^* = \lim \sup X = \inf \{v_m : m \geq 1\}$

If in a bounded real sequence $X : v_m =\sup \{x_n : n \geq m\}$, then : $x^* = \lim \sup X = \inf \{v_m : m \geq 1\}$ Proof Attempt : If $X= \{x_n\}$ is a sequence of real numbers which ...
0
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3answers
26 views

Bounded Sequences and Extrapolation of Convergence From Related Sequences

I'm considering some sequence $S_n$ which is bounded, and I want to prove that $S_n/n$ is convergent. I'm thinking that I could simply take $lim_{n \to \infty} S_n/n$ and simplify this to $(lim_{n \to ...
0
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1answer
13 views

Not existance of one sided limit

I have a question regarding one sided limits of functions. Let's say that the function $f$ is defined in $(a,b)$. And let's say that we want to check the limit of $f$ when it approaches b from the ...
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3answers
57 views

Explanation of Cauchy's root test / criterion

I've been studying some general stuff in convergence and I'm struggling with Cauchy's criterion for convergence of an infinite series. I've read in textbooks that it suggests that terms in their ...
3
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1answer
58 views

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? Is my solution correct?

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? I am confused because my friend insists the series converges conditionally. I think the series diverges. Here is my process and solution: ...
3
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2answers
74 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
0
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0answers
49 views

Convergence of series $\sum\limits_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{\sqrt{n}}$ and approximation with maximum error

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
0
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0answers
16 views

Prove that if $\{a_n\}$ converges, then $\{a_{2n+1}\}$ converges, using definition of convergence [duplicate]

Use the definition of convergence to prove that if the sequence $\{a_n\}$ converges, and $b_n = a_{2n+1}$, then the sequence $\{b_n\}$ also converges. Progress I was thinking about using proof by ...
0
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1answer
25 views

Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
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0answers
19 views

how to dertermine this series convergence?

Does the series (n-1)/ ( n sqrt(n)) converge from 1 to infinite. why? Do we have to use comparaison test? I don't try nothing because I don't know how to do that lol thanks
0
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1answer
14 views

Finding Convergeance sum for two power-series.

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
3
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0answers
93 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
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0answers
31 views

Ordinary Differential Equation with a trigonometric function: radius of convergence?

For the equation $$x^2y'' + y' + \tan(x)\,y = 0$$ establish lower bounds for the radius of convergence about the point $$x_0 = 1.$$
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4answers
102 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
0
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1answer
29 views

Distance between two compact subsets is always $\geq$ distance between two particular points of the subsets

Show that if $K_1$ and $K_2$ are compact subsets of $\mathbb R^p$, then there exist points $x_1$ in $K_1$ and $x_2$ in $K_2$ such that if $z_1$ belongs to $K_1$ and $z_2$ belongs to $K_2$, then ...
0
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1answer
34 views

Uniform convergence and relative error?

I never took a class where uniform convergence was introduced, so I only know the Definition and not much more about it. I have an Approximation of some sequence of functions $f_n$ by some function ...
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3answers
33 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
3
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1answer
31 views

Help Understanding Step in Proof of Convergence

The theorem is If $\sum a_n$ is a series of complex numbers which converges absolutely then every rearrangement of $\sum a_n$ converges, and they all converge to the same value. The proof ...
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1answer
58 views

Consider $f_n(x)={x^n-x^{3n}}$

A. For what values of x is the function series is point-wise convergent, and to what function? B. For what values of x is the series uniform convergence? My answers in the textbook are: A. As $n\to ...
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3answers
54 views

Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge?

The integral is $\int\left(\,1 + n^{2}\,\right)^{-1/4}\,{\rm d}n$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$ \sum\left(\, 1 ...
0
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4answers
63 views

Does the series $\sum \frac{1}{n\ (\ln(n))^{3/2}}$ converge or diverge?

Consider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but ...
1
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1answer
43 views

Will this series converge? If so, what is its limit?

If $a_n=(a_{n-1}+a_{n-2})/2$ and $a_1, a_2$ are given, will this series converge? And if so, what is the limit? By intuition I think it converges to $(a_1+2a_2)/3$ , but I am not able to prove it.
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1answer
41 views

what is the difference between bounded and convergent?

I know that bounded means to have an upper or lower bound. Let $E \subset \mathbb{R}$ be nonempty. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ ...
3
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1answer
35 views

Convergence of sequence of function

I need to check if sequence of functions $f_n(x):=\sqrt{x^2+\frac{1}{n}}$, $n\in \mathbb{N}$ converges (pointwisely, uniformly) in intervals:$[-1;1]$ and $\mathbb{R}$. Is there any algorithm how to ...
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2answers
34 views

What does tightness of convergence mean and why do we use this?

In a paper that I am reading it is written: We have $$(X(t_j))_{j=1}^k \xrightarrow{d} (Y(t_j))_{j=1}^k,$$ where $t_0=0, t_1 < \dotsb < t_k$. It further is written: In order to show that ...
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1answer
38 views

Convergence of proposed approximations to conditional expectation

Let $X,Y$ be real random variables on $(\Omega, \mathscr{F}, P)$ with $E[|X|] < \infty$. Let $Z_1, Z_2, \dots$ be a sequence of proposed approximations of $E[X|Y]$ defined by $$Z_n(\omega) = ...
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2answers
76 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
2
votes
2answers
42 views

Will the Newton's method be convergent to the root of the following function: $f(x)=\frac{-x}{x^2-1}$?

Will the Newton's method be convergent to the root of the following function, if the starting point $x_0>1$ will be chosen? $$ f(x)=\frac{-x}{x^2-1} $$
2
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0answers
16 views

scalar dimension to the approximation of an integrable function

Let $(M,\mathcal{A},\mu)$ space of probability. If $f\in L^1(\mu)$ then $f=\displaystyle{\lim_{n \rightarrow +\infty}\sum_{i=1}^n}\alpha_i\chi_{C_i}$ where $C_i\in \mathcal{A}$ and $\alpha_i \in ...
0
votes
1answer
27 views

Assumption in a convergence proof

Im in the middle of a proof of the fact that for a>0, if lim $x_n$ = a, then lim $\sqrt{x_n}$ = $\sqrt{a}$. I'm in the step that i use $| \sqrt{x_n} - \sqrt{a} |$ = $ \frac{|x_n - a|}{ ...