Convergence of sequences and different modes of convergence.

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2
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0answers
31 views

Convergence of a Markov Chain to the normal distribution

If $i$ is a state of an irreducible, postive recurrent Markov chain $X$, and $V_n$ is the number of visits to $i$ between times $1$ and $n$, and further $\mu=\mathbb{E}_i(T_i)$ and ...
2
votes
4answers
76 views

Convergence of $\sum_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$

I have encountered the following problem: Determine whether $$\sum \limits_{n=0}^{\infty} \left(\sqrt[3]{n^3+1} - n\right)$$ converges or diverges. What I have tried so far: Assume that $a_n = ...
2
votes
1answer
39 views

Prove that $\lim_{n\to \infty}\langle \operatorname{erfc}(-nx), \phi\rangle =\langle H_0, \phi\rangle $

Define the error function $\operatorname{erf}(x)$ as: \begin{equation} \operatorname{erf}(x):=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-y^2}dy \end{equation} and ...
2
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2answers
60 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead ...
3
votes
5answers
464 views

Why do some series converge and others diverge?

Why do some series converge and others diverge; what is the intuition behind this? For example, why does the harmonic series diverge, but the series concerning the Basel Problem converges? To ...
2
votes
1answer
337 views

how to show the convergence of an algorithm

I have two unknown variables x and y which are non linear equations to be solved. \begin{eqnarray} y=\frac {|\sin(2x+\theta)|}{\sin x\sqrt{A+2B\cos(2x+\theta)}} \nonumber \\ x=\arccos\bigg( ...
0
votes
0answers
15 views

uniform converging functions

If I know that $\sum_M^\infty v(x)$ converges towards $f$, then $$\left| \left( f + \sum_{1}^{M-1} v(x) \right) - \sum_{1}^N v(x)\, \right| = \left|\;f - \sum_{M}^N v(x)\,\right| < \epsilon$$ for ...
0
votes
2answers
52 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
0
votes
0answers
19 views

Is there something wrong with this proof? Convergence of m-tail of series

I wanted to formally proof that the uniform convergence of the $m$-tail of a series of functions implies uniform convergence of the entire series. It made intuitive sense but I did not know any ...
0
votes
1answer
31 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
3
votes
1answer
51 views

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
0
votes
2answers
34 views

When using the Integral test, why is the value of the integral different from the sum of the series?

According to my textbook, the value of the improper integral is not always equal to the sum of the series. But why is that?
-1
votes
3answers
65 views

Determine wether the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ diverges or converges? - I want to check if my reasoning is correct

I got that the series: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ converges by doing the following: $\sum_{n=0}^\infty\frac{n^22^{n+1}}{3^{n}}$ = $\sum_{n=0}^\infty ...
2
votes
2answers
23 views

Does one of these conditions on a sequence imply the other one?

Let ${(r_n)}_{n \geq 0}$ be a sequence of integers $\geq 2$. Set $q_n=\prod_{i=0}^{n-1} r_i$ (agreeing with $q_0=1$). I want to know whether one of these two conditions implies the other one (I think ...
-1
votes
2answers
67 views

How $(f_n)$ converges uniformly on $[a, b]$

Let $(f_n)$ be defined and continuous on an interval $[a, b]$, and differentiable on $(a, b)$. Let $c \in [a, b]$. Assume that $(f_n(c))$ converges and that $(f'_n)$ converges uniformly on $(a, b)$. ...
1
vote
1answer
22 views

Convergent Sequence and its limit

Can anybody help me out in this problem: I am not able to figure out how the value of lambda in the 2nd problem comes out to be 2 ? In the first problem value of lambda came out to be 2 after ...
4
votes
1answer
58 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
0
votes
2answers
26 views

Sequence of monotone functions problem

Found a problem on "The Way of Analysis" by Strichartz. Say $f_n$ converges to $f$ (pointwise) and each $f_n$ is increasing. (a) Must $f$ be increasing? (b) What happens if each $f_n$ is strictly ...
0
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0answers
20 views

Better chances on a gambling game [closed]

I'm working on a program that I hope it will help me to get better chances on a gambling game. I have a very large database of numbers how occur in this game and my program pass through those numbers ...
3
votes
1answer
34 views

Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
1
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1answer
36 views
2
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1answer
34 views

Uniform convergence for the sequence $\psi_n(x)=n^{-1}e^{-n^2/(n^2-x^2)}$

How can I prove that the sequence of functions: \begin{equation} \psi_n(x)=\begin{cases} n^{-1}e^{-n^2/(n^2-x^2)}, & |x|\leq n \\ 0, & |x|\geq n \end{cases} \end{equation} convergences ...
0
votes
4answers
56 views

Show that $\sum_{n=1}^{\infty} \dfrac{1}{n} \ln \left( 1+\dfrac{x}{n} \right)$, with $x>-1$ is pointwise convergent.

Title says it all, really. I have to show, that: $\sum_{n=1}^{\infty} \dfrac{1}{n} \ln \left( 1+\dfrac{x}{n} \right)$, with $x>-1$ is pointwise convergent, but I have no idea where to start. I am ...
10
votes
2answers
3k views

Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
0
votes
0answers
18 views

Expression about $\sum_{d=0}^{\infty} e^{dt}\frac{(5d)!}{(d!)^5}5\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right)$ [Done]

The purpose of my question in (here)[converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $ was actually to re-expression of following computation. ...
6
votes
1answer
39 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
0
votes
0answers
25 views

Prove that the given sequence is convergent

prove that the sequence $\frac{2n^2 + n + 10}{n^2 + 5}$ is convergent I've come down as far as $\left|\frac{n}{n^2 + 5}\right| < \epsilon$, however i don't know what the next step is.
0
votes
1answer
28 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
0
votes
2answers
27 views

Convergent series with condition

There exists a sequence of strictly decreasing real positive numbers $x_n$ such that its series converges, but the quantity $$\frac{x_n^2}{x_n-x_{n-1}}$$ doesn't converge to zero? All the famous ...
1
vote
2answers
46 views

Radius of convergence of $\sum_{n\geq 0}a_{n}x^{n}$.

Consider a series $\sum_{n\geq 0}a_{n}x^{n}$ where $a_{0}=2/3$ and $a_{n}=2-(1/2)a_{n-1}$ for all $n$. It is assumed that $2/3\leq a_{n}\leq 5/3$ for all $n\geq 1$. My problem is about determining its ...
0
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3answers
35 views

convergence of series $\sum_{n=1}^{\infty}\frac{4n^2+5n}{n(n^2+1)^{\frac{3}{2}}}$

I can't find a way to decide whether it convergent or not... tried the root test, but the result is 1(which means nothing)
-1
votes
2answers
40 views

for what valus of p, the series converge [closed]

For which values of $p>0$ does the series $\sum_{n=2}^\infty \frac{1}{n(\ln(n))^p}$ converge?
1
vote
1answer
17 views

Show that the radius of convergence of a sum of series is at least as big as minimum of radii of these series.

I am struggling with the following task. Suppose $\sum^{\infty}_{n=0}a_nx^n$ has radius of convergence $R$ and $\sum^{\infty}_{n=0}b_nx^n$ has radius of convergence $S$. I want to show that the ...
0
votes
2answers
59 views

A series with the recursive formula.

A sequence $\lbrace a_{n}\rbrace_{n\geq 0}$ is constructed by choosing a value of $a_{n}$, and then the following elements are determined from the equation $a_{n}=2-\frac{1}{2}a_{n-1}$ for ...
0
votes
0answers
11 views

Convergence in Distribution for two Dirichlet distributions

I'm working on a problem and I wanted to get some hints on how to solve it. To me, it seems like showing convergence to distribution but since it's been a while that I've not worked on these types of ...
0
votes
1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
8
votes
1answer
827 views

Proof of a theorem of Cauchy's on the convergence of an infinite product

Well it is relatively well known that the condition for absolute convergence is given by the following theorem: In order that the infinite product $\prod _{n=1}^{\infty }\left( 1+a_{n}\right) $ may be ...
2
votes
1answer
296 views
1
vote
1answer
28 views

Does the following sequence of random variables converge?

Let $X_1,X_2,...$ be independent random variables with $P[X_n=0]=1-1/n$, $P[X_n=1]=1/2n$, $P[X_n=-1]=1/2n$ Does $X_n$ converge almost surely? , Does $X_n$ converge in probability? I just started to ...
2
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1answer
48 views

Prove that $\sum _{ n=1 }^\infty \left[ \frac { p(p+1)\cdots(p+n-1) }{ q(q+1)\cdots(q+n-1) } \right] ^\alpha $ converges

i need to prove this $$\sum _{n=1}^\infty \left[ \frac { p(p+1)\cdots (p+n-1) }{ q(q+1)\cdots (q+n-1) } \right]^\alpha, \qquad (p>0,q>0)$$ converges if and only if $\alpha (q-p)>1$ I ...
0
votes
2answers
32 views

Series: only the latter terms matter

I've been told that when it comes to uniform convergence of series, only the tail matters, This seems intuitively obvious, but is there a theorem one can refer to? Further, if $\sum_{m}^\infty ...
13
votes
2answers
106 views

Suppose $\sum x_n$ converges (not necessarily absolutely), does $\sum \sin x_n$ necessarily converge?

The question is: If $\sum x_n$ converges, does $\sum \sin x_n$ converge? I know that if $\sum x_n$ converges absolutely, then $\sum \sin x_n$ converges. My intuition is that we cannot completely ...
0
votes
1answer
15 views

Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
5
votes
1answer
55 views

converge value of series $\sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) $

\begin{align} \sum_{n=0}^{\infty} \left( \frac{1}{n+d+1} - \frac{1}{n+5d+1} \right) = \sum_{n=0}^{\infty}\frac{4d}{(n+d+1)(n+5d+1)}= ? \end{align} I know from the $p$-test, ($i.e$ $\sum \frac{1}{n^p}$ ...
1
vote
0answers
24 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
-1
votes
0answers
47 views

Use the Lipschitz estimate to prove [closed]

Let ($f_n$) be a sequence of functions that are continuous on $[a, b]$ and differentiable on $(a, b)$. How to Use the Lipschitz estimate to prove that $|f_n(x) - f_p(x) - (f_n(c) - f_p(c))| \leq ...
0
votes
2answers
51 views

Prove the convergence of series $\sum_{k=1}^{\infty}\log(1+\frac{1}{\sqrt{k}})$ by Cauchy criterion

Given $$\sum_{k=1}^{\infty}\log\left(1+\frac{1}{\sqrt{k}}\right)$$ and by definition I need to prove that for $\forall \epsilon>0, \exists n_0 \text{ s.t. } \forall n>n_0, \forall p=1,2,...$ ...
2
votes
1answer
58 views

The series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$; the convergence and the differentiability.

Consider the series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$ for $x>-1$. a) Show that the series is pointwise convergent. Answer: I actually don't know how to show ...
1
vote
2answers
25 views

Bounding a summation by an integral

In progression of another question, my lecturer states $$\sum_{k=N+1}^\infty \frac{1}{k^2} \le \int^\infty_N \frac{1}{x^2}\, {\rm d}x.$$ However we have not covered bounding summations by integrals ...
0
votes
1answer
20 views

Does $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$ imply convergence of $\mathbf{x}^{(n)}$?

A sequence $\{\mathbf{x}^{(n)},n=1,2,...\}$. If $\lVert\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\rVert_2\rightarrow0$, does it also imply the convergence of the whole sequence $\{\mathbf{x}^{(n)}\}$?