Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

0
votes
0answers
42 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 ...
1
vote
0answers
21 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
votes
3answers
34 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 $\lim \sup (x_n) \leq \lim \sup (y_n)$ ...
1
vote
0answers
19 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
votes
3answers
15 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
votes
1answer
9 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 ...
1
vote
3answers
47 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
votes
1answer
51 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
votes
2answers
61 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
votes
0answers
43 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
votes
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
votes
1answer
22 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 $\mathcal{O}( h^\alpha )$ ...
0
votes
0answers
13 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
votes
1answer
12 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
votes
0answers
59 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 ...
1
vote
0answers
25 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 $$x_0 = 1.$$
1
vote
4answers
83 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
votes
1answer
23 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
votes
1answer
25 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 ...
1
vote
3answers
30 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
votes
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 ...
-3
votes
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 ...
0
votes
3answers
37 views

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

The integral is $\int { (1+n^2)^{-1/4}}dn$ is not quite possible, so I should make a comparison test. What is your suggestion? EDIT: And what about the series $$\sum (1+n^2)^{-1/4} \cos ...
0
votes
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
vote
1answer
41 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.
0
votes
1answer
34 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
votes
1answer
30 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 ...
-3
votes
0answers
20 views

Proving convergence using epsilon N definition [closed]

jesuserent answers (and they were negative), so I must've been doing it wrong or something.
1
vote
2answers
33 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 ...
1
vote
1answer
33 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) = ...
1
vote
2answers
72 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
41 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? $$ f(x)=\frac{-x}{x^2-1} $$
-3
votes
0answers
21 views

Is the series convergent or divergent [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum \sqrt{a_na_{n+1}}$ always convergent?Either prove it or give a counterexample.
1
vote
0answers
15 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
26 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|}{ ...
0
votes
2answers
41 views

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then it is true that $|x-c|< \epsilon$

If $x = lim (x_n)$ and if $|x_n - c| < \epsilon~~ \forall ~n \in N$, then is it true that $|x-c|< \epsilon?$ I am a little confused about this question ( it appears in the Bartle's elements of ...
1
vote
1answer
25 views

How does $\mathcal{L}^1$-convergence of a series of $\mathcal{L}^1$ random variables imply that $\sup_{n \in \mathbb{N}} \mathbb{E}[|X_n|] < \infty$?

Let $(X_n)_{n \in \mathbb{N}}$ be a series of random variables with $\forall i: X_i \in \mathcal{L}^1(\Omega, \mathfrak{F}, P)$ and $X_n \rightarrow^{\mathcal{L}^1}X$. How do I show then, that ...
0
votes
1answer
53 views

Find a Cauchy sequence that does not converge

I am supposed to look at $l_0$, the set of all sequences with finitely many non-real elements in $(l_0,d_{\infty})$. It is just that I don't quite understand how the $d_\infty$-metric is defined on ...
-1
votes
1answer
41 views

Proving $\lim_{n\rightarrow\infty} \sin(n)/n = 0$ using epsilon definition

So the limit as $n \rightarrow \infty$ of $2\sin(n) / n$ is $0$. How do I prove this? I say; $$|2\sin(n) / n - 0 | = |2\sin(n) / n| < \epsilon$$ for which $n$'s ? Not sure how to get rid of the ...
0
votes
1answer
16 views

Convergence of sequences and limits

Let $(f_n)$ be the fibonacci sequence and let $x_n = \dfrac{f_{n+1}}{f_n}$. Given that $\lim_{n \to \infty}(x_n) = L$ exists, determine the value of $L$.
2
votes
2answers
17 views

approximate a Borel set by a continuous

I wonder if it is possible to approximate a Borel set by a continuous function i.e. Let $B$ a Borel set in $(X,d)$ (compact separable metric space) I wonder if there continuous functions ...
2
votes
3answers
43 views

$\sum_{n=1}^{+\infty} \frac{n^{n-1}v^n}{n!}$ for what value of $v$ this series will be convergent? How to proceed for it?

I am interested in the convergence of the series $$\sum_{n=1}^{\infty} \left( \frac{n^{n-1}}{n!}v^{n} \right).$$ This series defines the tree function.
0
votes
1answer
34 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
1
vote
0answers
30 views

solution of the set of real non-linear equations

I have a set of real non-linear equations as following: \begin{equation} y_0 = f(y_0,y_1) \\ y_1 = g(y_0,y_1,y_2) \\ y_2 = g(y_1,y_2,y_3) \\ \vdots \\ y_{n-1} = g(y_{n-2},y_{n-1},y_n) \\ y_n = ...
1
vote
1answer
41 views

Does uniform convergence depends on the metric?

Definition: Let $f_n:X\to Y$ be a sequence of functions from a set $X$ to the metric space $Y$. Let $d$ be the metric for $Y$. The sequence $(f_n)$ d-converges uniformly to the function $f:X\to Y$ if ...
0
votes
0answers
36 views

proving convergence and finding the limit of this sequence if it is convergent?

$a(n)=\sin(1+a(n-1))$ where $a(0)=0$ and $n$ belongs to integers , prove or disprove whether the sequence converges and find the limit . well i understood that this is bounded but i am not able to ...
0
votes
2answers
30 views

A query in the proof of convergence of the set $\{1/n\}$

I have a query regarding the proof of the statement that the set $S = \left\{ \dfrac {1}{n} \right\}$ has limit point $0$. I am studying an introductory course in Analysis. Proof: From the ...
0
votes
1answer
44 views

Is this series divergent or convergent?

I've been stuck with this problem for a couple of days trying to solve it but got no where till now. The problem states that we have to prove if the series given below is convergent or divergent, if ...
0
votes
0answers
21 views

Sequence of Functions that does not Converge

I'm asked to show that the sequence of functions $f_n(x) = n^2x^n$ defined on the closed interval $[0,1]$ does not converge pointwise to any function as $n \to \infty$. For $0 \le x \lt 1$ I think I ...
0
votes
1answer
30 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...