Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

2
votes
2answers
31 views

AM-GM Inequality Confusing

Here is something that I find hard to make sense of. Suppose $X_1, X_2, ..., X_n$ are independent draws from some distribution. By AM-GM inequality, we have: $$ \left( X_1 X_2 .. X_n ...
0
votes
1answer
19 views

Proving that the sequence converges

I would like some help with the following problem. Thanks for any help in advance. Let $(x_n)$ and $(y_n)$ be convergent sequences of positive real numbers. Let $ x_n \xrightarrow[n \to \infty]{} x$ ...
3
votes
2answers
89 views
+50

Can you get a closed-form for $\prod_{p\text{ prime}}\left(\frac{p+1}{p-1}\right)^{\frac{1}{p}}$?

When I use the Taylor expansion series for $$\log(1+x)^{1+x}+\log(1-x)^{1-x}$$ with $x=\frac{1}{p}$, $p$ prime, I believe that I can deduce $$\sum_{p\text{ ...
3
votes
0answers
47 views

Checking whether sequence $ x_n = \ln(n^2 + 1) - \ln(n) $ converges or diverges

I have to show whether $$ x_n = \ln(n^2 + 1) - \ln(n) $$ converges or diverges. I can write $$ x_n = \ln(n^2 + 1) - \ln(n) = \ln\left(\frac{n^2+1}{n}\right) = \ln\left(n + \frac{1}{n}\right). $$ ...
1
vote
1answer
25 views

Pointwise or Uniform convergence?

Consider the functions $f_n:[-1,1]\to\mathbb{R}$ defined by $$f_n(x):= \frac{x}{\sqrt{x^2 + \tfrac 1n}}$$ and determine whether the convergence is uniform or pointwise. I can see that this will ...
0
votes
1answer
55 views

Prove that the series $\sum_{k=0}^\infty (-1)^k\ \left(\sqrt{k+1}-\sqrt{k}\right)$ converges but not absolutely.

I have to prove that the following series converges but not absolutely: $$\sum_{k=0}^\infty (-1)^k\ \left(\sqrt{k+1}-\sqrt{k}\right)$$ I have used the Leibniz test (alternating series test) to prove ...
1
vote
4answers
154 views

continuous map of metric spaces and compactness

Let $f:X\rightarrow Y$ be a continuous map of metric spaces. Show that if $A\subseteq X$ is compact, then $f(A)\subseteq Y$ is compact. I am using this theorem: If $A\subseteq X$ is sequentially ...
8
votes
8answers
5k views

Series that converge to $\pi$ quickly

I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does ...
1
vote
1answer
178 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
0
votes
1answer
133 views

Distribution in Polya's Urn / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
5
votes
3answers
145 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $ if \ 0 < |x-a| < \delta$ Question: Why can't we weaken the ...
1
vote
1answer
44 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
0
votes
2answers
34 views

How to prove an inifnite sum

How can I prove that $$\sum_{n=0}^{\infty}\frac{(-2)^n}{n!}=e^{-2}$$ I do know that $$\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^{x}=\lim_{n\rightarrow\infty}(1+\frac{x}{n})^n$$ But I never had a real ...
0
votes
1answer
18 views

$X$ sequentally Compact implies that $X$ is complete

I was reading through Roydens book and there is one part that I don't understand. Here is the proof. Suppose $X$ is sequentially compact metric space, then $$X \text{ is sequentally compact}:= ...
0
votes
1answer
61 views

convergence of $f(x+y)$ to $f(x)$ for $y >$ to $0$ in $L^1$.

I have to show the following: for $f \in L^1(S^1)$ Show that the map $f(x) \to f_y = f(x+y)$ is continuous in the distance of $L^1(S^1)$. i.e. $\lim_{y \to 0} ||f_y-f||_1 = 0$ I am supposed to use ...
0
votes
0answers
24 views

Absolute Convergence of Infinite Weierstrass Product

I am really stuck on something. I need to show the following: Let $U$ be a domain in $\mathbb{C}$. If $f_n: U \to \mathbb{D}$ are analytic functions satisfying that $\sum |f_n - 1|$ converges ...
0
votes
3answers
27 views

How do I decide which convergence criterium to use and what to do if they don't work?

Take $$ \sum_{n=1}^\infty (-1)^nn^{1/n} $$ I just blindly tried (Ratio test) $$ \limsup_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| $$ and (Root test) $$ \limsup_{k\to\infty} \sqrt[k]{|a_k|} $$ ...
1
vote
1answer
14 views

Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$.

Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$. Here's my ...
1
vote
3answers
24 views

Uniform convergence with two limits

I'm doing a question investigating uniform convergence of a function and I need something cleared up if possible. $f_n(x) = \frac{x^n}{1+x^n}$ on the interval $[0,1]$. Now, pointwise, this turns ...
0
votes
0answers
8 views

Square wave in the limit of infinite frequency

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
0
votes
1answer
2k views

Convergence and accumulation points proof.

Suppose $\{a_n\}_{n=1}^\infty$, converges to $A$ and $\{a_n: n \in J\}$ is an infinite set. Show that $A$ is an accumulation point of $\{a_n: n \in J\}$. So far I have done the basics of convergence ...
0
votes
0answers
14 views

Is $X_{k+1}=\frac{1}{N}\sum_{i=1}^N \Pi_{X_{k}^{1/2}v_i}$ globally convergent?

Let $X_0=X_0^\top\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix, let $v_i\in\mathbb{R}^n$, $i=1,\dots, n$, be a set of $n$-dimensional real vectors and pick an integer $N>0$. I ...
7
votes
2answers
494 views

Does strong law of large numbers hold for an average of this triangular array containing “almost i.i.d.” sequences?

If a sequence of $n$ i.i.d. positive random variables $X_1,\ldots,X_n$ has the expectation $\mathbb{E}[X_1]=\mu<\infty$, then it is known that the strong law of large numbers (SLLN) holds: ...
-2
votes
0answers
37 views

Series converges point-wise [on hold]

$$f_{n}=\sum_{n=1}^{\infty }\frac{x^{4}}{(1+x^{4})^{n}}$$ Show that it converges point-wise on $\mathbb{R}$, but not uniformly on $\mathbb{R}$. My attempt: I think, we should use Weierstrass's M ...
0
votes
1answer
28 views

Domain of $\lim_{n\to\infty}|(n+1)x| < 1$

In the process of doing the ratio test (for testing convergence of a function), I have the following issue. I am trying to find the convergence domain of $x$ in the following function: $$ ...
2
votes
2answers
29 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
1
vote
1answer
359 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
0
votes
1answer
28 views

Show the convergence of a sequence using a convergent sequence in a metric space

Let $X$ a metric space with metric $d$ and let $P \subset X$ not empty. Let $f :X \to \mathbb{R}$ a function and define $f(x) = \inf\{d(x,y): y \in P\}$. Let $(x_n)_{n \in \mathbb{N}}$ a convergent ...
-2
votes
0answers
19 views

Bounded, monotone, convergent [on hold]

Study if the sequence $X_n$ is bounded, monotone and convergent. If the sequence is convergent, find also its limit. $X_1 \in (0, 1), X_{n+1}=2X_n +13,n \in N$
4
votes
1answer
3k views

Bounded sequence and every convergent subsequence converges to L

Let $\{x_n\}$ be a bounded sequence such that every convergent subsequence converges to $L$. Prove that $$\lim_{n\to\infty}x_n = L.$$ The following is my proof. Please let me know what you think. ...
1
vote
2answers
49 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
0
votes
1answer
58 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...
0
votes
1answer
28 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
0
votes
1answer
28 views

use L1-convergence to show integral convergence

Let $f\in L^1([0,1])$, $g_n$ a sequence of continuous functions that converges in $L^1$ to some $g\in L^1([0,1])$. Now my question is: Does $\int_0^1 f(t)e^{g_n(t)} dt$ converge to $\int_0^1 ...
0
votes
0answers
18 views

Non-linear simultaneous recurrence system

Given a non-linear, non homogeneous, discrete time recurrence system: $a_i^t = f_i(a_1^{(t-1)},a_2^{(t-1)},\ldots,a_k^{(t-1)},C_1)$, for all $i\in [k]$ where $C_1,\ldots,C_k$ are constants and each ...
0
votes
0answers
24 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
0
votes
0answers
49 views

If $u \in W^{1,p}(I) \bigcap C_c(I)$ then $u \in W_{0}^{1,p}(I) $ where $ W^{1,p}(I)$ is the Sobolev Space

I want to show the following statement: If $u \in W^{1,p}(I) \cap C_c(I)$ then $u \in W_{0}^{1,p}(I) $. $W^{1,p}(I) $ is the Sobolev Space, i.e. the space consisting of the functions that are ...
1
vote
2answers
35 views

Convergence in $L^p$: $E[X 1_A] = E[X_n 1_A]$

Let $p>1$ and suppose that $X_n \rightarrow X$ in $L^p$ as $n \rightarrow \infty$. For $A \in \mathcal{F}_n=\sigma(X_0, \dotsc, X_n)$ it is written $$E[X 1_A] = E[X_n 1_A]$$ Can you explain me ...
2
votes
1answer
67 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
0
votes
0answers
50 views

Region of Convergence

I know to find the Region of Convergence you find the poles of the denominator, but I'm unsure what to do for the case where there is no denominator (for a and d) and what to do if the poles are ...
0
votes
0answers
19 views

Norm convergence of a net of operators

Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let ...
2
votes
1answer
25 views

Does $\mathbb{P}$-a.s. convergence preserve independence?

Let $\mathcal F$ be a $\sigma$-algebra and $X_n$ RV s.t. $X_n$ is independent of $\mathcal F$ for all $n$. Also let $X_n \to X$ $\mathbb{P}-$a.s.. Is $X$ independent of $\mathcal F$ now too?
2
votes
1answer
30 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
1
vote
1answer
35 views

An example in Cantor's intersection theorem if the hypothesis $\text{diam}(D_n)\to0$ as $n\to\infty$ is omitted

Cantor's intersection Theorem: If $(D_n)_{n=1}^\infty$ is a sequence of nonempty closed sets in a complete metric space $(X,d)$ such that $D_{n+1}\subset D_n$ for all $n\in\mathbb{N}$ and ...
0
votes
2answers
37 views

Weak convergence in Banach space

If $X$ is a Banach space, $X'$ its dual and $x,x_n\in X$ and $x',x_n'\in X'$, then the following implication holds: $x_n\to_w x$ (weak convergence) and $x_n'\to x'$ in $X'$ $\implies$ ...
6
votes
0answers
67 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges: $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...
-1
votes
1answer
47 views

Convergence of the method :Newton-Raphson.(GATE) [closed]

This question was asked in GATE exam and It's answer is 1. I could not understand this question. Suppose that the Newton-Raphson method is applied to the equation $2x^2+1-e^{x^2}=0$ with an initial ...
3
votes
1answer
460 views

Uniform convergence of $f_n = (n^a x^2)/(n^2 +x^3)$

My question is, if you have the sequence $$f_n = \frac{n^\alpha x^2}{n^2 +x^3}$$ on $[0, \infty)$, for values of a for $0<\alpha<2$ does the sequence uniformly converge? I guess another way to ...
0
votes
0answers
26 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
1
vote
5answers
42 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...