Convergence Definition and 1000 Threads

Homework Statement Let X_1, X_2... be a sequence of independent random variables with E(X_i)=\mu_i and (1/n)\sum(\mu_i)\rightarrow\mu Show that \overline{X}\rightarrow\mu in probability. Homework Equations NA The Attempt at a Solution I feel as if this shouldn't be too hard...

[FONT=trebuchet ms]Question from my homework sheet. Can someone confirm I've got these correct. Let (an)n∈N be any sequence of real numbers. Which of the following statements are true? Give precise references to the results in the Lecture Notes for those which are true. Construct counter...

(Hey guys and gals!) Homework Statement Given a bounded set x_n and for any y_n the following condition holds: \limsup_{n \rightarrow ∞}(x_n+y_n) = \limsup(x_n)+\limsup(y_n) Show that x_n converges. Homework Equations Definition of limsup(x_n) = L: \forall \epsilon > 0 \mid...

Once upon a time there was a boy, neigh a man! He had trouble understand the connection between continuity and the different test for convergence. Sadly, he seen that they were connected and started to study, yet to no avail. Can someone please lend a helping hand on this quest for adventure...

Hi Guys, I 'm currently trying to make a 2D model of a sector of a compressor disk with the blade attached to it by means of a frictional contact (as shown in the attached pic). The contact between the blade and disk is frictional (coeff=0.25), augmented lagrange formulation, and adjust to...

Hi, I try to prove, that function f_n = \frac{\sin{nx}}{\pi x} converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if f_n satisfy some conditions. So I need to show, that \lim_{n\rightarrow...

How to prove the convergence or divergence of ? $$\sum^{\infty}_{n=1}\frac{\Gamma(n+\frac{1}{2})}{n\Gamma{(n+\frac{1}{4})}}$$

Homework Statement Let \bar{X_n} denote the mean of a random sample of size n from a distribution that has pdf f(x) = e^{-x}, 0<x<\infty, zero elsewhere. a) Show that the mgf of Y_n=\sqrt{n}(\bar{X_n}-1) is M_{Y_n}(t) = [e^{t/\sqrt{n}} - (t/\sqrt{n})e^{t/\sqrt{n}}]^{-n}, t < \sqrt{n} b) Find...

I'd really appreciate some help with a sum of: a_n= |sin n| / n All I've thought of, is that I should probably create a subsequence of {a_n}, such that all the elements of this subsequence {a_n_k} are >epsilon >0, and then compare the subsequence to 1/n which diverges. However, I have no...

Homework Statement Hi all, I am back with more questions. Thank you to those helped with my last assignment. Question 1: For |x| ≤ π, define a sequence of functions by: Kn(x) = {n if -π/n ≤ x ≤ π/n, 0 otherwise} for natural numbers n. An earlier part of the question asked that I...

Homework Statement Show that the infinite series \sum_{n=0}^{\infty} (\sqrt{n^a+1}-\sqrt{n^a}) Converges for a>2 and diverges for x =2The Attempt at a SolutionI'm reviewing series, which I studied a certain time ago and picking some questions at random, I can't solve this one. I tried every...

im having trouble with this question - http://i.imgur.com/Ars4J1b.png - more specifically with part a, as i have a good idea how to go about b. given the initial value problem y' = 1-t+y , y(t0)=y0 show that the exact solution is y=\phi(t)=(y0-t0)et-t0+t we've only spoken of...

got Fourier series as a result of solving a PDE. how do i evaluate the converg. using average error in order to determine the # of terms needed for it to converge to less than X%?

In the page that I attached, it says "...while at the continuity points x of F_x (i.e., x \not= 0), lim F_{X_n}(x) = F_X(x)." But we know that the graph of F_X(x) is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of...

I was a bit confused with the pages that I attached... 1) "An intuitive estimate of \theta is the maximum of the sample". But we are only taking random samples, so even the maximum might be far from \theta, right? 2) I don't understand how E(Y_n) = (n/(n+1))\theta. I thought that E(Y_n) =...

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

Show that the sum of (n/(3n+1))n from n=1 to ∞ converges. The book solves this with a comparison test to (1/3)n, but I'm making a mistake with an n-th term test somewhere. an = (n/(3n+1))n Take ln of both sides, then use n = 1/(1/n) to setup for l'Hopital's rule. ln an = ln(n/(3n+1)) /...

I have one series \sum_{n=13}^{\infty}(-1)^{\left\lfloor\frac{n}{13}\right\rfloor} \frac{ \ln(n) }{n \ln(\ln(n)) } . How to investigate its convergence? I wanted to group the terms of this series but I don't know whether it's a good idea as we have 13 terms with minus and then 13 with plus and...

Homework Statement Suppose c_n is the digit in the nth place of the decimal expansion of 2^1/2. Prove that the radius of convergence of \sum{c_n x^n} is equal to 1. Homework Equations The Attempt at a Solution What I want to show is that limsup |c_n|^1/n = 1. Clearly for any...

It is well known due to the famous argument by Dyson that the perturbation series for quantum electrodynamics has zero radius of convergence. Dysons argument essentially goes like that: If the power series in α had a finite (r>0) radius of convergence it also would converge for some small...

Could anyone help me out with the monoticity of this sequence please? \frac{2\ln(n)}{\sqrt{n+1}} . It should decline. I am investigating the convergence of one series and I need it to do the Leibniz test.

I have a problem with convergence of two series: 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+... 1+ \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{1}{...

Def. Let $\{z_j\}$ be a sequence of non-zero complex numbers. We call the exponent of convergence of the sequence the positive number $b$, if it exists, $$b=inf\{\rho >0 :\sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\rho}}<\infty \}$$ Now consider the function $$f(z)=e^{e^z}-1$$ Find the zeros $\{z_j\}$...

A complex series need not be defined for all z within the "circle of convergence"? The (complex) radius of convergence represents the radius of the circle (centered at the center of the series) in which a complex series converges. Also, a theorem states that a (termwise) differentiated...

Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence: 1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...

1. Determine the raius of convergence and interval of convergence of the power series \sum from n=1 to \infty (3+(-1)n)nxn. 2. Usually when finding the radius of convergence of a power series I start off by using the ratio test: limn\rightarrow∞|((3+(-1)n+1)n+1xn+1/ (3+(-1)n)nxn| But...

I have a non-increasing sequence of random variables \{Y_n\} which is bounded below by a constant c, \forall \omega \in \Omega. i.e \forall \omega \in \Omega, Y_n \geq c, \forall n. Is it true that the sequence will converge to c almost surely? Thanks

Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...

\text{We need to prove that the sequence} \ a_{n} = \{n^{2}/2^{n}\} \ \text{converges to 0} \\ \text{Consider the sequence {n/ 2n} = { 0, 1/2, 1/2, 3/8, 1/4, 5/32, ...}. The terms get smaller and smaller.}\\ \\ \text{we can easily show that} \ n/2^{n}<=1/n \ \forall n>3 \\ \text{from the fact...

Homework Statement it should be all right this time, but could you please check my solution? prove the convergence and find the limit of the following sequence: ##a_1>0## ##a_{n+1}= 6 \frac{1+a_n}{7+a_n}## with ## n \in \mathbb{N}^*## The Attempt at a Solution the sequence is...

Homework Statement For question 25.15 in this link: http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw9sum06.pdf I have some questions about pointwise convergence and uniform convergence... Homework Equations The Attempt at a Solution Our textbook says...

Homework Statement Determine whether the following functional series is pointwise and/or uniformly convergent: \sum_{n=1}^\infty \frac{x}{n} (x\in\mathbb{R}) Homework Equations The Attempt at a Solution My answer to this seems very straightforward and I would be very grateful if...

Homework Statement Use the limit comparison test to determine whether the following series converges or diverges. \sum{\frac{2^n}{3^n - 1}} Where the sum is from n = 1 to n = ∞. Homework Equations The limit comparison test: Suppose an>0 and bn>0 for all n. If the limit of an/bn=c, where...

Sequence Convergence \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)} I have tried some comparisons bot not conclude: \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)}<=\sum_{n=2}^\infty\frac{(-1)^n}{1} \sum_{n=2}^\infty\frac{(-1)^n}{x-1}<=\sum_{n=2}^\infty\frac{(-1)^n}{ln(n)} Somebody having any insights...

Homework Statement let ##a_n## be ##a_{n+1}=\frac{1}{4-3a_n} \quad n≥1## for which values of ##a_1## does the sequence converge? which is the limit? The Attempt at a Solution ##0<a_1<\frac{4}{3}## because if ##a_1>\frac{4}{3}→a_2<0## not possible. Now let's assume ##a_n## converges to M. I...

Hi, I was playing with Riemann zeta function on mathematica. I encountered with a quite interesting result. I iterated Riemann zeta function for zero. (e.g Zeta...[Zeta[Zeta[0]]]...] It converges into a specific number which is -0.295905. Also for any negative values of Zeta function, iteration...

Homework Statement Ʃ (from n=1 to ∞) (4x-1)^2n / (n^2) Find the radius and interval of convergenceThe Attempt at a Solution I managed to do the ratio test and get to this point: | (4x-1)^2 |< 1 But now what? How do you get the radius and interval? Any help will be appreciated! Thanks

I think the solution I've found makes sense, but I'd like it to be double-checked. Homework Statement Let ##(a_n)## be a limited sequence and ##(b_n)## such that ##0≤b_n≤ \frac{1}{2} B_{n-1} ## Prove that if ##a_{n+1} \ge a_{n} -b_{n}## Then ##\lim_{n\to \infty}a_n## exists...

Homework Statement Find the radius and interval of convergence for the two series: 1) [(n+1)/n]^n * (x^n), series starting at n=1. 2) ln(n)(x^n), series starting at n=1. Homework Equations You're usually supposed to root or ratio your way through these. The Attempt at a...

Working out of Boas' Mathematical Methods in the Physical Sciences; Chapter 14, section 2, problem 42... I'm supposed to write the power series of the following function, then find the disk of convergence for the series. Boas goes on to state, "What you are looking for is the point nearest...

ƩHomework Statement Determine the radius of convergence and the interval of convergence for the follwing function expanded about the point a=2. f(x)= ln(3-x) Homework Equations ln(1-x) = Ʃ (x^n+1)/n+1 n=0 which has radius of convergence at |x|<1 The Attempt at a Solution...

$\sum\limits_{n = 2}^{\infty}\frac{1}{(\ln n)^{\ln n}}$ I am trying to show that this series diverges or converges

Homework Statement Hello noble physicists, I am struggling to solve a problem with any sort of confidence whatsoever, so to you I turn in the hopes of guidance. The problem refers to GRACE twin satellite convergence due to gravitational anomalies. I’d like to estimate the...

Hello, I am trying to prove/disprove the following claims: (1) There exists a sequence a_n of positive number so that the series Ʃ a_n converges whereas the series Ʃ (a_n)^2 diverges. I believe I managed to disprove that. Is it indeed false? (2) There exists a sequence of real...

Hi, How may I know whether the series ((-1)^n)[cos (3^n)x]^3/(3^n) converges/diverges?Should I use the Leibniz Criterion? It is stated that (cos a)^3 = (1/4)(3cos a + cos 3a)

Homework Statement Use any method or test to see if the series converges or diverges. Homework Equations The series: ((1/n) - (1/n^2))^n The Attempt at a Solution Well the integral test won't work because there's no real integral for that according to Wolfram Alpha. Also if you...

f is a continuous function on [0,infinity) such that 0<=f(x)<=Cx^(-1-p) where C,p >0 f_k(x) = kf(kx) I want to show that lim k->infinity ∫from 0 to 1 of f_k(x) dx exists so my idea is if I have that f_k(x) converges to f(x)=0 uniformly which I was able to show and that f_k(x) are all...

Homework Statement Determine whether the following series diverges, converges conditionally, or converges absolutely. \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) Homework Equations The Attempt at a Solution This was on today's test, and was the only problem I wasn't able to solve. I doubt my...

Hello all, this came up in my chemistry class when our prof. showed a graph of the size of atomic orbitals (or orbital energy) in relation to the Z number (or number of protons). He did this to show that as the z number increases the size of the orbitals also decrease (because effective nuclear...

The question is: Show that if \suman from n=1 to ∞ converges absolutely, then \suman2 from n=1 to converges absolutely. I'm not sure which approach to take with this. I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or...