Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

2
votes
2answers
76 views

Do the ratios of successive primes converge to a value less than 1?

I think it's a pretty straightforward question. Does $\lim_{n \to \infty}{\frac{p_n}{p_{n+1}}} < 1?$ ***$p_n$ denotes the nth prime. Since the average gap increases between successive primes by ...
1
vote
2answers
57 views

How to prove or disprove the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n\cdot a_{n}}{n}$?

Question: Assume that $a_{n}\in\mathbb R$, and let the series $$\sum_{n=1}^{\infty}a_{n}$$ be convergent I would like to prove or disprove the convergence of ...
3
votes
2answers
75 views

Prove that if an infinite series converges, then the associative property holds

I'm self-studying from the book Understanding Analysis by Stephen Abbott and have no idea how to do exercise 2.5.2 on page 57. The exercise is as follows: Prove that if an infinite series converges, ...
0
votes
1answer
31 views

Criteria for measure convergence implying convergence a.e.

Suppose the function $g_n = \sup_{m \geq n} |f_n-f_m|\to 0$ in measure. Show $f_n \to f$ a.e. Suppose instead that $\sum_{n=1}^{\infty} m\{ |f_n - f|>\epsilon\} < \infty$. Show $f_n \to f$ a.e. ...
0
votes
1answer
67 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
0
votes
1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
0
votes
1answer
23 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
0
votes
1answer
48 views

$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$

$$\lim_{x\rightarrow {0}^{+}}\sum_{n=1}^{\infty}{(-1)}^{n-1}\frac{1}{n!{x}^{n}}=?$$ I found this question during my study. In my opinion ,it is not difficult to solve ,but it is interesting. So I ...
1
vote
1answer
35 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
2
votes
2answers
46 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
4
votes
5answers
173 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
0
votes
1answer
32 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
1
vote
1answer
49 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
0
votes
0answers
19 views

Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
1
vote
1answer
185 views

How to find a series for comparison with $\sum 1/\sqrt{n(n+1)}$?

The series $$\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n(n+1)}}$$ I have tried ratio and integral both lead me to inconclusive, so probably it's by comparisson but I can't find What to compare.
3
votes
1answer
35 views

If $\sum_{n \geq 1}X_n$ converges a.s. then $\forall a > 0: \sum P(|X_n|>a) < \infty$

I'd like to show that for $(X_n)_{n\geq 1}$ a sequence of real-valued and independent random variables, If $\sum_{n \geq 1}X_n$ converges a.s., then $\forall a > 0: \sum P(|X_n|>a) < ...
3
votes
2answers
69 views

$\lim \sqrt[n]{a^n + b^n}$

I've seen some answers here for why this limit is the maximum between $a$ and $b$, but all of then included the hypotesis that $a$ and $b$ are both non negative. It was asked to show that this limix ...
2
votes
2answers
64 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
3
votes
2answers
68 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
0
votes
0answers
15 views

Laurent Series for $\frac{1}{e^z-1}$-Radius

Can someone provide the radius of convergence of the Laurent Series of $\frac{1}{e^z-1}$ on $\mathbb{C} \backslash 0$? In class we reduced it to an inequality, but there seemed to be some debate ...
1
vote
2answers
36 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
0
votes
1answer
29 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
1
vote
2answers
25 views

Limiting variable in interval: Lebesgue Dominated Convergence

So I am pretty comfortable using the LDCT for definite integrals and summations, but I am looking at a problem that has the interval as a function of the limiting variable, i.e.: $$\lim_{n\to\infty} ...
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
1
vote
0answers
47 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
0
votes
1answer
44 views

Convergence of function

Suppose that $V(t)$ is nonnegative continuous function ($\forall t:V(t)\ge0$). $\dot{V}(t) = -|h(t)|^2 + f(t)g(t)$ $f(t)$ is a bounded and uniformly continuous function. $g(t)$ is a bounded and ...
0
votes
1answer
27 views

Show convergence in mean

Suppose $X(t)$ is a positive i.i.d. random process in discrete time with finite mean $a$. Let $Y(t)=\min \{a, X(t)\}$. Question is: (1)Can we claim that ...
1
vote
1answer
25 views

Bounded Almost Sure convergence implies convergence in pth mean

A book I'm reading gave the following result. If $X_n \to X $ a.s. and $|X_n|^p \le Z$ for some random variable $Z$ with finite expectation, then we have convergence in $p$th mean. I was wondering, if ...
1
vote
1answer
34 views

Hilbert Space: Weak Convergence implies Strong Convergence

This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ...
1
vote
2answers
65 views

Convergence of infinite $ \sum (\frac{n}{n+1})^{n^2} $?

I have being trying to solve this convergence but with no success. Using the ratio test I have reached here: $$ a_n = \left( \frac{n}{n+1} \right)^{n^2} $$ And $$ \frac{1}{a_n} = \left( ...
2
votes
1answer
48 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
2
votes
2answers
57 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
5
votes
1answer
42 views

Show that $C^1([0,1])$ is not reflexive

Aim of this exercise is proving that $(C^1([0,1]),\|\cdot\|_{C^1})$ is not reflexive. We know that, if $(f_h)_h\subset C^1([0,1])$ is a sequence that weakly converges to $f\in C^1([0,1])$ (that is ...
2
votes
0answers
40 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
0
votes
2answers
61 views

Converges Or Diverges: $\sum _{n=1}^{\infty }\:e^{-\sqrt{n}}$

Converges Or Diverges: Attempt: int_1^∞ e^(-sqrt(n)) dn t = -sqrt(n); dt = -dn/(2*sqrt(n)); int -2*sqrt(n)/[-2*sqrt(n)] * e^(-sqrt(n)) dn int (2t)*(e^t) dt u = 2t; du = 2 dt; dv = e^t dt; ...
0
votes
1answer
52 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
2
votes
2answers
60 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
-4
votes
2answers
100 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
2
votes
1answer
40 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
2
votes
1answer
35 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...
4
votes
2answers
92 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
4
votes
3answers
126 views

$\lim_{x\rightarrow 1}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^x}=\ln2$.

Prove $$\lim_{x\rightarrow 1}\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\ln2.$$ Of course $$\sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}}=\ln2,$$ but we can not use the Proposition : If a ...
2
votes
1answer
41 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
0
votes
1answer
111 views

Gradient descent (with line search) for convex functions viewed as alternation

I have fundamental confusion about gradient descent (with line search) and the reason it works. I try to explain my view here, and please tell me where it goes wrong. Let $f: \mathbb{R}^n \to ...
3
votes
1answer
84 views

Compute a sequence $A_n$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$

How can we compute a sequence $A_n$ of positive real numbers, such that $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{A_n\ln(A_n)}=1$? One way I can think of, is by defining $A_n\ln(A_n)=2^n$, but ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
53 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
0
votes
0answers
41 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
0
votes
2answers
53 views

Finding the convergence radius of $\dfrac{(n!)^k\cdot x^n}{(kn)!}$

If K is a integer positive, find the convergence radius of the series $$\sum\limits_{n=0}^{\infty} \dfrac{(n!)^k\cdot x^n}{(kn)!}$$ Any initial idea?
2
votes
2answers
27 views

Convergence of $\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$

This question is a place to store proofs of the convergence of Euler's product formula for the gamma function: $$\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$$ which is convergent for ...