Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

4
votes
3answers
76 views

Does the integral converge? $\int_1^\infty \frac{ln(1+x)}{x^2}dx$

Does the integral converge? $$\int_1^\infty \frac{ln(1+x)}{x^2}dx$$ Well, I used integration by parts and got to $ln4$, which means it is clearly converges. but I want to try another approach as this ...
3
votes
1answer
106 views

Two nondecreasing sequences that bound each other

Question: Let ($a_n$) and ($b_n$) be two nondecreasing sequences with the property that, for each positive integer $n$, there are integers $p$ and $q$ such that $a_n \leq b_p$ and $b_n \leq a_q$. ...
5
votes
1answer
59 views

Use of directed sets in the definition of nets in topology

In topology, we use nets instead of sequences. The motivation is quite natural since the sequence is not "long" enough if the neighborhoods of some point "separate" too much. What I am confused ...
1
vote
6answers
53 views

Supremum proof simple

I got stuck on this problem and can't figure it out, I hope somebody can help me, I also wrote my attempt. Thanks in advance!! Question: Let $(a_n)$ be a convergent sequence in $\mathbb{R}$. $a_n ...
0
votes
1answer
20 views

Correct method of Proving Raabe's test?

I was wondering if my method of proof for Raabe's test was valid, since it is different from the normal method used with comparing to a sequence $\frac{1}{n^{p}}$ for some p > 1. Raabe's Test (As ...
1
vote
2answers
53 views

Which metric to use to make the sequence 1, 1.4, 1.414, 1.4142, .. converges in space Q?

In space Q, with the metric it inherits from R, the sequence 1, 1.4, 1.414, 1.4142, ... does not converge. Is there a way to change the metric to make it converge in Q?
1
vote
1answer
25 views

Non-linear systems convergence

Is there a way of being sure that simple iteration schemes, such as Gauss-Jacobi and Gauss-Seidel will converge for non-linear systems? I understand that for linear systems, the matrix A has to be ...
2
votes
1answer
40 views

Prove convergence of: $ \sum_{n=1}^\infty\frac{(-1)^n\cdot\sqrt{n}}{(n+1)\cdot2^n}\cdot(x-3)^n $

I would like to prove the convergence of series: $$ \sum_{n=1}^\infty\frac{(-1)^n\cdot\sqrt{n}}{(n+1)\cdot2^n}\cdot(x-3)^n $$ for every x $\in \mathbb{R}$. I am a bit lost on this one. I tried using ...
1
vote
0answers
43 views

Convergence of a sum of random variables with Bernoulli coefficient.

I present a problem which is connected with some of my previous questions. Suppose that $Y_t$ is a "regular enogh" (for example $Y_t=W_t$ with $W_t$ a Brownian motion) stochastic process with ...
1
vote
1answer
56 views

What about $\lim_{x\to 1}\left(\zeta(x)-\frac{1}{x^x-1}\right)=1+\gamma$?

When I type 1 in the box lim x to, and zeta(x)-1/(x^x-1) in the box Function, of this online calculator (Wolfram Alpha) one has as output $$\lim_{x\to ...
1
vote
1answer
25 views

Implications of convergence in probability

Consider two sequences of real-valued random variables $\{X_n\}_n$, $\{Z_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose that (1) $Z_n\in o_p(1)$, i.e. $Z_n$ ...
1
vote
0answers
16 views

Convergence of a sequence of Bernoulli variables.

For $\lambda\in(0,1)$ consider the following sequence of Bernoulli random variables $$ \mathbb{P}\left[B_n=1\right]=1-\frac{\lambda}{n},\quad \mathbb{P}\left[B_n=0\right]=\frac{\lambda}{n}. $$ Now ...
0
votes
1answer
41 views

Do these infinite series converge to a finite limit? [duplicate]

Now, I know that there is that remarkable result which finds that $$\sum_{n=1}^{\infty}n=-\frac{1}{12}$$ for $n\in\mathbb{N}$, under some kind of Cauchy limit. Are there any such convergences for ...
0
votes
0answers
26 views

Is the below assertion true? Why?

Let $n\in N$, $a_n$ be a real sequence such that $|a_{n+1}-a_n|\rightarrow 0$ as $n\rightarrow \infty$, then $a_n$ is convergent in $R$.
1
vote
0answers
36 views

Every nondecreasing function on [0,1] is the pointwise limit of a sequence of continuous functions.

Prove:Every nondecreasing function on $[0,1]$ is the pointwise limit of a sequence of continuous functions. I know every nondecreasing function can only have at most countable discontinuous point, ...
1
vote
1answer
22 views

Is this a sufficient condition for a.e. convergence?

Suppose one has a sequence $(f_{n})_{n \in \mathbf{N}}$ of real-valued, non-negative functions defined on a finite measure space $(X, \mu)$, with the following property: For every $n \in ...
1
vote
0answers
31 views

Determine if this power series is convergent for Z on the boundary of it's disc of convergence

So, this is the power series: $\sum_{n=1}^\infty$$(\frac{1+i\sqrt 3}{n^2})$$((\frac{z}{2})+i)^n$ I have already found the centre power series to be $C_n$= $-2i$ And the Radius of Convergence using ...
2
votes
3answers
46 views

Does $\sum_{n=1}^\infty\frac{\cos\left(\left(2n-1\right)x\right)}{2n-1}$ converge absolutely

$$\sum_{n=1}^\infty\frac{\cos\left(\left(2n-1\right)x\right)}{2n-1}$$ We have that $\sum_{n=1}^\infty|\frac{\cos\left(\left(2n-1\right)x\right)}{2n-1}| \leq \sum_{n=1}^\infty \frac{1}{2n-1}$. However ...
0
votes
1answer
25 views

little o notation in equations

Consider the real-valued functions $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$. Suppose I have an equation saying $$ f(x)=g(x)+o(1) \tag{$*$}$$ as $x \rightarrow ...
3
votes
6answers
131 views

Does $\sum_{k=1}^{\infty}\frac{k!}{k^k}$ converge?

I have tried using ratio test: $$P =\lim_{k\rightarrow\infty}\left|\frac{(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{k!}\right|$$ $$ P=\lim_{k\rightarrow\infty}\left|\frac{(k+1)\cdot ...
1
vote
1answer
20 views

Does $\frac{1}{n}x_0$ converge to the origin in any topological vector-space?

Let $X$ be a topological $\mathbf{R}$-vector-space (not necessarily Hausdorff) and $x_0 \neq 0$ a non-zero element of $X$. Then intuitively the sequence $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
0
votes
1answer
13 views

Newton method norm of error is proportional to norm of residual?

Let $F(x):\mathbb{R}^n\rightarrow \mathbb{R}^n$. Newton's method is: $x_{k+1} := x_k + d_k$, where $d_k$ is computed to satisfy $F'(x_k)d_k = -F(x_k)$. If the error at the current step is $e_k = x^* ...
2
votes
0answers
38 views

Measure Theory - understanding usage of Dominated Convergence Theorem?

I am trying to understand the given proof below. But I don't really understand how the Theorem of Dominated convergence is applied? Which is the function that "dominates" the sequence? and we need ...
0
votes
0answers
49 views

Convergence in $L^{1}$ of martingale

I have problems with the following task: Let $X_{n} $ for $n=1,2,3...$ be independent random variables with distribution $B(n,\frac{1}{n})$ and define $Y_n=X_1...X_n$. Is $Y_n$ convergent in $L^1$? ...
1
vote
3answers
79 views

Does the sequence $\sum \limits_{n=1}^{\infty} \frac{\log n}{n^2}$ converge absolutely?

Does the sequence $\sum \limits_{n=1}^{\infty} \frac{\log n}{n^2}$ converge absolutely? I know that $\sum \limits_{n=1}^{\infty} \frac{1}{n^s}$ for $s >1 $ converges absolute. So is it possible ...
2
votes
0answers
26 views

Convergence of infinite-dimensional random variables

For every $n \in \mathbb{N}$ and every measurable $E \subseteq [0,1]$, the object $f_n(E)$ is a random variable that takes real values. The sequence ($f_n$) can thus be understood as a sequence of ...
3
votes
1answer
70 views

Characterisation of convergence of bounded sequences via ultra-filters

Let $\{a_n\}_{n\in\mathbb N}$ be a bounded sequence of real or complex numbers and $\mathscr F\subset\mathscr P(\mathbb N)$ be a non-principal ultra-filter. Then $a=\lim_{\mathscr F}a_n$ is ...
1
vote
1answer
32 views

Properties of S_n = 1 + 1/2 + … + 1/n

Let $S_n=\sum_{k=1}^n \frac{1}{k}$. Are the following correct? $S_{2^n}\geq \frac{n}{2}$ $\frac{S_n}{n}\to 1$ as $n\to \infty$. $S_{2^n}=\sum_{k=1}^{2^n} \dfrac{1}{k}$. Proceeding by induction ...
2
votes
2answers
29 views

limit and convergence of a summation

I have two related questions here: Known: $q$ is a positive integer $s=\frac{3}{\alpha}+\epsilon$ $k$ is sufficiently large to ensure $\frac{2\sqrt{2}}{k^{\alpha}} \le \delta$ $\alpha > 3$ ...
0
votes
1answer
25 views

Type of convergence of a Cauchy sequence of functions on a complete metric space?

Let $\{f_n\}$ be a Cauchy sequence of functions defined on a complete metric space $E$. Then $f_n \to f$ on $E$. What is the type of this convergence? Is it pointwise?
0
votes
0answers
34 views

Rate of Convergence of $x_{k+1} = \frac 14 (x_k^2 +sin(x_k) + 1)$

I have shown that the fix point Iteration $$x_{k+1} = \frac 14 (x_k^2 +\sin(x_k) + 1)$$ has exactly one fix point $x^*\in[0,1]$ for every $x_0 \in [0,1]$ with the Banach theorem. Now I want to find ...
0
votes
1answer
47 views

Showing that, for all polynomial $p(x)$ with $\deg(p)>1$ the series $\sum 1/p(n)$ converges.

How can I show that, for all polynomial $p(x)$ with $\deg(p)>1$, the series $\sum 1/p(n)$ converges? I tried comparison, but it works only for polynomials in $\mathbb{N}[x]$.
0
votes
1answer
36 views

For which values of $\rho$ does the CES production function satisfy the Inada conditions

Given $F$ is a constant elasticity of substitution (CES) production function: $$F(K,AL) = \left [ \alpha K^{\rho} + (1-\alpha) (AL)^{\rho} \right ]^{\frac{1}{\rho}},$$ where $\alpha \in \left ( 0,1 ...
1
vote
0answers
17 views

Uniform convergeness and polynomials

If $f(x)$ is continuous function on $[a,b]\Rightarrow\exists <P_{n}(x)>$ sequence of polynomials such that $P_{n}(x)$ uniformly converges to $f(x)$ on $[a,b]$. Is this statement true or false? ...
2
votes
3answers
44 views

Is $f_{n}(x)=x^{n}$ uniformly convergent on the interval $[0,1]$?

Let $f_{n}(x)=x^{n}$. I know that $f_{n}(x)$ converges to $0$ on the interval $[0,1)$ and converges to $1$ on $x=1$. But is $f_{n}(x)=x^{n}$ uniformly convergent on the interval $[0,1]$?
0
votes
3answers
89 views

Showing that the series $\sum \log{n}/n^2$ converges.

I aim to show that the series $$\sum \frac{\log{n}}{n^2}$$ converges. I know that the inequality $log(n) < \sqrt{n}$ holds for large $n$. So this give us one way to prove the convergence of $\sum ...
0
votes
1answer
38 views

Checking whether this sequence is convergent or divergent

Q: Check if the following series is convergent or divergent and if convergent, find the limit: $$\frac{1+\frac{1}{2}x+\frac{1}{3}x^\frac{1}{2}+\frac{1}{4}x^\frac{1}{3}+\frac{1}{5}x^\frac{1}{4}+ ...
1
vote
0answers
16 views

From deterministic little o to stochastic little o notation

I have a question on Lemma 2.12 in van der Vaart which can be found here. Consider statement (i): if $R(h)\in o(||h||^p)$ as $h\rightarrow 0$ $\Rightarrow $ $R(X_n)\in o_p(||X_n||^p)$. Among the ...
2
votes
1answer
40 views

convergence of continued nested function

Does $$\sin (x + \sin (x + \sin (x + \sin (x + ... ) ) ))) $$ converge to any limit? If so, to what? Recursive plotting appears at times to come to same/similar profiles.
-2
votes
1answer
73 views

Prove that $f_{n}(x) = \frac{1}{x} \chi _{[\frac 1 n, 2]}$ is not uniformly convergent on $[0,2]$?

Let the sequence of functions $\{f_{n}\}_{n=1}^{\infty}$ be defined by $f_{n}:[0,2] \to \mathbb{R}$ where $f_{n}(x) = \left\{ \begin{eqnarray} 0 &,& 0 \le x < \frac 1 n \\ \frac 1 x ...
1
vote
3answers
114 views

Show $S_n = \frac{n}{2n+1}$ converges to $\frac{1}{2}$

This is my thinking so far but it is by no means a rigorous proof: $\frac{n}{2n+1} < \frac{n}{2n} = \frac{1}{2}$ I don't know how to translate this result into a proof, thank you.
-1
votes
0answers
21 views

Boundedness of expectation

Suppose I have a sequence of random vectors $\{X_n\}_n$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, where $X_n:\Omega \rightarrow \mathbb{R}^h$. Each $X_n$ induces the ...
2
votes
1answer
81 views

If $\det{(H_{p}^{\infty})}$ converges to a constant value, estimate the range of $p$.

Introduction: One day I calculated the value of determinant which is like Hilbert matrix $H_{p}^{n} \in \bf{R}^{\it{n \times n}}$using my computer. The determinant is defined below. $$ ...
3
votes
4answers
100 views

The sequence $\phi_n= 1+\frac{1}{2^4}+\frac{1}{3^4}+\ldots+\frac{1}{n^4}$ is bounded above .

A sequence $(\phi_n)$ is defined as follows : $$\phi_n= 1+\frac{1}{2^4}+\frac{1}{3^4}+\ldots+\frac{1}{n^4}$$ Show that the sequence is convergent. Because this sequence is monotonic, proving it is ...
0
votes
0answers
31 views

2-Dimensions Integral convergence

Does the following integral converge on $\mathbb{R}^2$: $\int \int \frac{log(x^2+y^2)}{x^2+y^2}dxdy$ I found that inside the unit circle the integral is -$\infty$ and outside the unit circle its ...
7
votes
1answer
129 views

Why does this double infinite sum converge to $e$?

I can't seem to come to grips with the result below: $$S=\sum_{n=1}^\infty \sum_{k=n}^\infty\frac{1}{k!}=e$$ which is given by Mathematica (code below) and (numerically) verified by WolframAlpha. ...
1
vote
2answers
25 views

Equivalence of limit notation

Consider a sequence $\{a_n\}_n$ with $a_n \in \mathbb{R}$ $\forall n$. We know that the following expression are equivalent: $$\lim_{n \rightarrow \infty} a_n=a \tag{$*$}$$ $$\lim_{n \rightarrow ...
1
vote
1answer
29 views

Construction of a sequence + conditional convergence

EDIT: I figured out a sequence that works. Thanks all. Have a question regarding Real Analysis. I've worked hard trying to come up with a sequence which satisfies the question, but I'm not having any ...
1
vote
1answer
37 views

prove $\frac{n e^{-n^2 x^2}}{\sqrt{\pi }}$ converges to $\delta(x)$

How can show as $n$ goes toward infinity the sequence converges to $\delta(x)$ my problem is I don't know how to show this.
0
votes
4answers
98 views

why is $\lim\limits_{k\to\infty}\frac{1+k}{k^k}=0?$

My question is, why is $$\lim\limits_{k\to\infty}\frac{1+k}{k^k}=0?$$ I tried to prove it with L'Hospital's rule: $\lim\limits_{k\to\infty}\frac{1+k}{k^k}=\lim\limits_{k\to\infty}\frac{1}{kk^{k-1}}$ ...