Convergence of sequences and different modes of convergence.

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1answer
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From pointwise convergence to uniform convergence

Consider a real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $a\in \mathbb{R}$. Let $\{f_n(X, a)\}_n$ be a sequence of real-valued random ...
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2answers
46 views

Solving these types of integrals, using Monotone convergence theorem and Dominated convergence theorem.

I'm allowed to use these two theories and obviously the standard techniques when solving integrals. $$\lim_{n\to \infty } \int_{0}^{1}\frac{n^{\frac{1}{2}} x \ln x}{1+n^2x^2}dx$$ I did a similar ...
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3answers
65 views

Sum of a certain series [closed]

I'am having problems here. How to find the sum of this series? The factoral is confusing me. $$\sum_{n=2}^{\infty} \frac{(-1)^{n-1} 2^{n-1}}{(n-1)!}$$
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3answers
227 views

Can epsilon be a matrix?

Question In the following expression can $\epsilon$ be a matrix? $$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + ...
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3answers
53 views

Prove that $\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$ converges [closed]

Prove that the following power series converges: $$\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$$ I have tried using d'Alembert's ratio test however this was inconclusive. Anyone have any ideas?
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0answers
21 views

Is $\sum \frac{n+1}{b_{n+1}}$ irrational, when $b_1=2$ and $b_{k+1}=2^kb_k(b_k-1)+1$, $k\geq 1$?

Let the sequences of positive integers $$a_n=n$$ when $n\geq 1$, and $$b_{n+1}=2^nb_n(b_n-1)+1$$ for $n\geq 1$ taking $b_1=2$. I've computed with previous sequences to assert that satisfy the ...
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1answer
44 views

Study the convergence of this series of functions: $\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$

I tried to study the convergence of this series: $$\sum_{n=1}^{\infty }n^x\left ( \tan\frac{x^n}{n}-\sin\frac{x^n}{n} \right )$$ I started to study the pointwise convergence with the limit for $n ...
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0answers
16 views

proof of a convergent subrow in every row in B using diagonal argument

full question I got a hint that for the compactness of M I need to show that every row in B has a convergent subrow (diagonal argument) but I don't know how to show this, does anybody know how to ...
2
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1answer
37 views

Series expansions with alternating series from endpoints: conjecture

A student of mine made the following two conjectures after working through examples of checking endpoints of series expansions, and although I feel that the conjectures are both false, I have not come ...
0
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1answer
27 views

Power series confusion when multiplying fractions.

I am stuck on the following question. check that the following sum from 0 to infinity converges using power series. sum of $$ 1/((n+(1/2))^2)$$ the next line of work is : $$4/((2n+1)^2)$$ I have ...
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2answers
32 views

Problems using D'alemberts Ratio test for convergence or divergence

The geometric series is as follows : $$n/2^n$$ I am using the ratio test therefore comparing : $$(n+1/2^{n+1})/ (n/2^n)$$ my next line of work is : $$(n+1/2^{n+1}) * (2^n/n)$$ however I am not ...
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0answers
30 views

Random Samples and Sample Variance Bound

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population. Show that: $$\max_{1 \leq i \leq n}|X_{i}-\bar{X}|<\frac{(n-1)}{\sqrt{n}}S$$ Where we have the sample variance ...
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1answer
20 views

Example of absolute convergent series divergent [duplicate]

I have learnt that in complete norm space any series that is absolute convergent is convergent. However, I am wondering is there any example of divergent series which is absolute convergent in that ...
2
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1answer
43 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
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5answers
60 views

Finding $\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$

Finding $$\sum_{k=1}^{\infty}k^2 \frac{2^{k-1}}{3^k}$$ I think if $$\int_{1}^{\infty}x^2 \frac{2^{x-1}}{3^x}dx$$ exists that this sequence is convergent, but I doubt that this integral is equal to ...
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3answers
31 views

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$

Prove wether or not the following series diverges or converges: $\sum_{n=0}^\infty {(-1)^nn\over n+1}$ I am just not sure, I know if I use the absolute value test for convergence and root test it ...
1
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0answers
40 views

Define a sequence by two conditions and show this sequence converges and find its limit

Question: Define a sequence $(a_n)$ by two conditions $(a_1) = \sqrt{2}$ and $(a_n+1)= \sqrt {2+(a_n)}$ for $n \ge 1$. Show that this sequence converges and find its limit. (Suggestion: Show that the ...
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3answers
75 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
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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 $b_n \leq a_q$. ...
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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 ...
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6answers
52 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 ...
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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 ...
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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?
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1answer
22 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
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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 ...
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0answers
41 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 ...
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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 ...
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1answer
23 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. $Z_n$ ...
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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
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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 ...
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0answers
25 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 $R$.
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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, ...
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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 ...
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0answers
30 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 ...
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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 ...
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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 ...
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6answers
130 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 ...
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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 $(\frac{1}{n} x_0)_{n \in \mathbf{N}}$ ...
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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^* ...
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0answers
37 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 ...
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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 $L^1$? ...
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3answers
77 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 ...
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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
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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 ...
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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
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2answers
28 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
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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?
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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
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1answer
46 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
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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 ...