Convergence Definition and 1000 Threads

Homework Statement Hi, everyone. I'd appreciate it if someone could explain something for me regarding the convergence of series. Thanks in advance![/B] Homework Equations In my calculus book, I'm given the following: (1) - For p > 1, the sum from n=1 to infinity of n^-p converges. (2) -...

Hello, Two questions will be posed here. (1) Question about Convergence; quick way. Hello, I am trying to learn this concept on my own. My major question here is that, Is there a quick way, to tell if an integral converges or diverges? Suppose $\int_{0}^{\infty} \frac{x^3}{(x^2 +...

Hello Normally in order to change the order of limit and integration in rimann integration, you need uniform convergence. But let's say that you are not able to prove uniform convergence, but only pointwise convergence. And let's say you are able to prove that the functions are also...

Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$

HI guys,this is my first programming experience , i have developed an MATLAB code for steady state heat conduction equation , on governing equation dt2 /dx2 + dt2/dy2 = -Q(x,y) i have solved this equation with finite difference method, As far as i know if we increase the mesh size it leads...

Homework Statement I've been given a question that makes use of 5^(n)*sin(pi*n!) The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity? So basically, I...

Homework Statement Show that if given \mathbf{x}_0, and a matrix R with spectral radius \rho(R)\geq 1, there exist iterations of the form, \mathbf{x}_{n+1}=R\mathbf{x}_0+\mathbf{c} which do not converge. The Attempt at a Solution Let \mathbf{x}_0 be given, and let...

We work on project about mobile cellular networks. In a part of the project, we faced the problem about the proof of convergence of ...(Complete description is Available in attachment.Please download it) Please help me.It is very important!

1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...

my final is tomorrow and my instructor gave a list of questions that will be similar to the ones on the final exam and i want to see how they should be done properly. I've been working on other problems but i can't get past these ones. thanks determine convergence/divergence...

find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?

find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$

I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States: " If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...

Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it. We have the equation \begin{equation} \partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right) \end{equation} We will look for solutions satisfying...

[SIZE="4"]Definition/Summary In what follows, we will work in a normed space (X,\|~\|). A series is, by definition, two sequences (u_n)_n and (s_n)_n such that s_n=\sum_{k=0}^n{u_k} for every n. We call the elements u_n the terms of the series. The elements s_n are called the partial...

Hello. How do I find the radius of convergence for this problem? ##\alpha## is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$ I understand that we can use the ratio test to find R. And by using ratio test, I got R=1. But in the...

Hello. How do I find the radius of convergence for this problem? $\alpha$ is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$

Hello. I am stuck on this question. I'd appreciate if anyone could help me on how to do this. The question: Expand the following into maclaurin series and find its radius of convergence. $$\frac{2-z}{(1-z)^2}$$ I know that we can use geometric series as geometric series is generally...

on page 4, example 9 in this link, http://www.personal.psu.edu/auw4/M401-notes1.pdf, they show a sequence of functions is not uniformly convergent. To show this, you need to show that for some epsilon, there is no 'universal' N. But they didn't pick a particular value of $z$, they chose...

Hello. I need explanation on why the answer for this problem is R = ∞. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$...

Hello. I need explanation on why the answer for this problem is $R=\infty$. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$ Divide...

Just a few quick questions this time: I'm doubting the first one mostly, because when I used the integral test to evaluate it: I ended up getting (-1/x)(lnx +1) from 2 to infinity, which gave me an odd expression: (-1/infinity)(infinity +1 -ln2 -1). I'm assuming this means it is convergent...

Hey guys, I have a couple more questions. For the first one, taking the limit to infinity obviously equals 0 so it should be convergent, right? Also, for the second one, the limit as n approaches infinity for gives me indeterminate form, so I took the derivative which just gave me ln(n)...

Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm highly doubtful of my answer for c. I used the roots test instead of the ratio test, which gives 1/n, which I took the limit of to get an interval of [-∞ , ∞] As for a and b, I got [-5,5] and (-∞, ∞)...

Hey guys, I have a few more questions for the problem set I'm working on at the moment: I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute...

Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm mostly unsure of my response for b. For a, I just split the series into two parts and added 6+3 to get 9, and thus the series is convergent. For c, I got 3/5 after taking the limit, which is...

This thread is only for question 5. As for number 5 part a, after tediously expanding the partial fraction expression, I ended up getting c=1, d=0, b=1, and c=1, ultimately resulting in: ln(x) - (1/x^2) + c. I really don't think this looks right. As for 5b, I obtained b=-1, c=-1, a=2, and...

For a,b, and c respectively, I got divergent (to -infinity), convergent (to π/6), and divergent (to infinity, since the first part's sum is 1/3, but lim negative infinity gives infinity, thus the summation of the two integrals gives a divergent integral). I'm sure these are right, but I'd...

Hello, I'm doubting a couple of my answers for these questions. Some of them seem relatively simple, but there are slight nuances that I'm not sure of. This thread is only for question 4. For 4a, I just used the (a^2) - (x^2) => x=asin(Ø) rule and substituted 3sin(Ø) for x. I ended up...

$$(x-1)-\frac{(x-1)^2}{2!}+\frac{(x-1)^3}{3!}-\frac{(x-1)^4}{4!}+ ∙ ∙ ∙$$ well this looks like an alternating-series, the question is: at what value(s) of x does this converge. one observation is that if x=0 then all terms are 0 so there is no convergence, also I presume you can rewrite this...

Homework Statement Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally? \sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n}) Homework Equations Alternating series test The Attempt at a Solution I started out by taking the limit of ##a_n...

Homework Statement Let ##X## be a topological space. Let ##A_1 \supseteq A_2 \supseteq A_3...## be a sequence of closed subsets of ##X##. Suppose that ##a_i \in Ai## for all ##i## and that ##a_i \rightarrow b##. Prove that ##b \in \cap A_i##. Homework Equations The Attempt at a Solution...

Homework Statement For which integer values of p does the following series converge: \sum_{n=|p|}^{∞}{2^{pn} (n+p)! \over(n+p)^n} Homework Equations The Attempt at a Solution I'm trying to apply the generalised ratio test but get down to this stage where I'm not sure what...

Homework Statement Does the following series converge or diverge? ##∑\frac{n^5}{n^n}## (as n begins from 1 and approaches infinity) Homework Equations Ratio test? The Attempt at a Solution For your reference, thus far I have learned about the geometric series, the limit test...

Homework Statement For 0<q<∞, and x rational, for what x values does the series converge? \sum_{n=0}^{∞} q^{1/n} x^nThe Attempt at a Solution I don't know which method works best for this

Homework Statement Given a non-negative sequence \{a_{n}\}_{n=1}^{\infty}. Proove that the serie \Sigma_{n=1}^{\infty}a_{n} converge if and only if \Sigma_{n=1}^{\infty}\ln(1+a_{n}) converges. Homework Equations The Attempt at a Solution My first attempt is the direct...

Homework Statement Check if the series below converge. a) $$\sum_{n = 1}^\infty \frac{n}{2n^2 - 1}$$ b) $$\sum_{n = 2}^\infty (-1)^n \frac{2n}{n^2 - 1}$$ Homework Equations The Attempt at a Solution For a). The series converge if the sum comes up to a finite value. If...

I'm looking for the proof of the following theorem/statement. $$ \begin{align} \lim_{n \to \infty} M_{X_n}(t) = M_X(t) \end{align} $$ for every fixed t \in \mathbb{R}. My book only states the theorem without proving it and I haven't found a proof online. Any help is appreciated! :)

Here is the question: I have posted a link there to this thread so the OP can view my work.

Assume the sequence of positive numbers ${a_n}$ converges to L. Prove that $\lim_{n \to \infty} \sqrt[n]{a_1a_2...a_n} = L$ (The nth root of the product of the first n terms) Since ${a_n}$ converges we know that for every $\epsilon> 0$ there is an $N$ such that for all $n > N$ $ |a_n -...

Define ##\rho(f)=\int |f|\mathrm d\mu## for all integrable ##f:X\to\mathbb C##. This ##\rho## is a seminorm, not a norm. Does ##\rho(f_n-f)\to 0## imply ##f_n\to f## a.e.? I kind of think that it should, because in the case of real-valued functions, ##\int|f_n-f|\mathrm d\mu## is the area...

Homework Statement Let f be a non-negative measurable function. Prove that \lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.The Attempt at a Solution I feel like I'm supposed to use the monotone convergence theorem. I don't know if I'm on the right track but I created a...

Hi, I am writing to you to ask for help. I have a problem with the contact "Frictional". I'll show you screenshots of my settings and program the console errors. Everything will be shown in the screenshots. The problem is that between the beams and the top element and the bottom element is...

Homework Statement I know that the harmonic series is divergent.But why \frac{-1}{n} is also divergent? I've search for some test to test that, but I could not find a method for negative series. So how can I prove the series is divergent?:confused: Homework Equations The Attempt...

Homework Statement Series: \sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n} what is the behaviour of the series at radius of convergence \rho_o=-z ? Homework Equations The Attempt at a Solution So I can specify that the series is monatonic if z is non negative as...

Homework Statement determine the radius of convergence of the series expansion of log(a + x) around x = 0 Homework Equations The Attempt at a Solution So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the...

Homework Statement I've found that the typical way for using ratio test is to find the limit of an+1/an However, my tutor said that radius of convergence can be found by finding the limit of an/an+1 and the x term is excluded. For example:Finding the interval of convergence of n!xn/nn my...

I am currently learning series and testing for convergence. For comparison tests especially I am having an issue grasping the concept of picking a proper limit to compare too. For example the following problem If someone could please put it in the form where it actually looks like what it...

Homework Statement Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges. The Attempt at a Solution I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!

a) Show that sum_(n=0)^infinity (2^n x^n)/((1+x^2)^n) converges for all x in R\{-1,1} b) Even though this is not a power series show that sum above = 1 + sum_(n=1)^infinity (2nx^n) for all -1<x<=1. For part a by the ratio and root test we get |(2x)/(1+x^2)| but this does not have an n in it...