Convergence Definition and 1000 Threads

Homework Statement This is for Calculus II. We've just started the chapter on Infinite Series. n runs from 1 to ∞. \Sigma\frac{1}{n(n+3)} The Attempt at a Solution I used partial fraction decomposition to rewrite the sum. \frac{1}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}...

Determine whether the sequence converges or diverges, if it converges fidn the limit. $$a_n = n \sin(1/n)$$ so Can I just do this: $$n * \sin(1/n)$$ is indeterminate form so i can use lopitals so: $$1 * \cos(1/x) = 1 * 1 = 1$$ converges to 1?

Determine whether the sequence Converges or Diverges. Tricky question, so check it out. $$\frac{n^3}{n + 1}$$ So here is what I did divided out n to get $$\frac{n^2}{1} = \infty \therefore$$ diverges Now, here is what someone else did. They applied L'Hopitals, and then claimed that...

Determine whether the series is convergent or divergent. $$\sum^{\infty}_{n = 1} \frac{n - 1}{3n - 1}$$ I ended up with $$\frac{1}{3} * 1 = \frac{1}{3}$$ , which is 0.333 ... so wouldn't that mean that $$r < 1$$? Also wouldn't that mean that it is convergent since $$r < 1$$ ? I don't...

Homework Statement Consider the sequence {an}\subsetR which is recursively defined by an+1=f(an). Prove that if there is some L\inR and a 0≤c<1 such that |\frac{a_{n+1}-L}{a_{n}-L}|<c for all n\inN then limn\rightarrow∞an=L. Homework Equations Definition of convergence: Suppose (X,d) is...

Homework Statement infinity ∑ (x^(n+5))/3n! Find the radius of convergence and interval of convergence n=0 Homework Equations The Attempt at a Solution I got 0 for the radius and 0 for the interval of convergence using the ratio test. This is no right. Can someone please help me? Thank you

hey pf! if we have a piecewise-smooth function ##f(x)## and we create a Fourier series ##f_n(x)## for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus ##\lim_{n \rightarrow \infty} f_n(x) \neq f(x)##? thanks!

Homework Statement Show that if a>-1 and b>a+1 then the following integral is convergent: ∫(x^a)/(1+x^b) from 0 to ∞ The Attempt at a Solution x^-1 < x^a < x^a+1 < x^b x^-1/(1+x^b) < x^a/(1+x^b) < x^a+1/(1+x^b) < x^b/(1+x^b) I also know any integral of the form ∫1/x^p...

Homework Statement Define f_n : \mathbb{R} \rightarrow \mathbb{R} by f_n(x) = \left( x^2 + \dfrac{1}{n} \right)^{\frac{1}{2}} Show that f_n(x) \rightarrow |x| converges uniformly on compact subsets of \mathbb{R} Show that the convergence is uniform in all of \mathbb{R}...

Homework Statement Test to see if \sum_{n=1}^{\infty}{ i^n/n } converges.Homework Equations See above. The Attempt at a Solution If I separate this series into real/imag. parts both series diverges by the integral test. However, according to Wolfram Alpha, the series converges to...

Homework Statement Prove for c>0 the sequence {x_n} = \frac{1}{2}(x_{n-1} + \frac{c}{x_{n-1}}) converges. The Attempt at a Solution This is proving difficult, I have never dealt with recursive sequences before. Any help would be appreciated. Thanks.

Consider the signal \[ x(t) = e^{-5t}\mathcal{U}(t) + e^{-\beta t}\mathcal{U}(t) \] and denote the its Laplace transform by \(X(s)\). What are the constraints placed on the real and...

Homework Statement Show that the sequence {(p_{n})}^{∞}_{n=0}=10^{-2^{n}} converges quadratically to 0. Homework Equations \stackrel{limit}{_{n→∞}}\frac{|p_{n+1}-p|}{|p_{n}-p|^{α}}=λ where α is order of convergence; α=1 implies linear convergence, α=2 implies quadratic convergence, and so...

Determine whether the integral is Divergent or Convergent$$\int^0_{-\infty} \frac{1}{3 - 4x} dx$$ I did a u substitution and got $$\lim_{a\to\infty} -\frac{1}{4}\sqrt{3} + \frac{1}{4}\sqrt{3 - 4a}$$ So is because the $$-\infty$$ is under the square root is it going to be divergent? I have...

Homework Statement Show that {n2 - n + 5} is increasing and hence show that {xn} is convergent when {xn} = exp[(3n2 - 3n +14) / (n2 - n + 5)] You may assume exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3. Homework Equations The Attempt...

A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I...

Homework Statement Use $$\sum\limits_{n=1}^∞ \frac{1}{n^2}$$ to prove by the comparison test that $$\sum\limits_{n=1}^∞ \frac{n+1}{n^3} $$ converges.Homework Equations $$\sum\limits_{n=1}^∞ \frac{n+1}{n^3} \equiv \sum\limits_{n=1}^∞ \frac{1}{n^2} + \sum\limits_{n=1}^∞ \frac{1}{n^3} $$ The...

Homework Statement Find the range of uniform convergence for the following series η(x) = ∑(-1)n-1/nx ζ(x) = ∑1/nx with n ranging from n=1 to n=∞ for both Homework Equations To be honest I'm stumped with where to begin altogether. In my text, I'm given the criteria for uniform...

Homework Statement Check whether the sequence a_{1}=\alpha ,\alpha > 0, a_{n+1}=6*\frac{a_{n}+1}{a_{n}+7} converges and find its limit if it does, depending on α. Homework Equations The Attempt at a Solution I showed boundedness([0,6]) and found that in the case of convergence...

Hi, I have to prove the following theorem: Let $f_n:[0,1] \to \mathbb{R}, \forall n \geq 1$ and suppose that $\{f_n|n \in \mathbb{N}\}$ is equicontinuous. If $f_n \to f$ pointwise then $f_n \to f$ uniformly. Before I start the proof I'll put the definitions here: $f_n \to f$ pointwise if and...

Let R be a sigma algebra and let $f$ be a real value function on R such that for a sequence ($A_{n}$) of disjoint members of R, we have that the sum of $f$($A_{n}$) over all n is equal to the image of the countable union under $f$. Prove that the sum of $f$($A_{n}$) is in fact absolutely...

Does pointwise convergence mean that |$f_{n}$-$f$|->0 for each x?

Homework Statement serie ((-1)^(n-1)(x-2)^(n-1))/(5^n) Homework Equations how to find the interval of convergence for this? The Attempt at a Solution

Homework Statement Let X_n \in Ge(\lambda/(n+\lambda)) \lambda>0. (geometric distribution) Show that \frac{X_n}{n} converges in distribution to Exp(\frac{1}{\lambda}) Homework Equations I was wondering if some kind of law is required to use here, but I don't know what Does anyone know how this...

Hi, I am trying to model a simple plane stress problem using Ansys. I am using Ansys 14.0. The problem is a simple square plate, without a corner, and with a hexagon hole around the midle. The boundary conditions consist of a constant pressure on the top side, and full constrain on the...

Homework Statement study the pointwise and the uniform convergence of ##f_{n1}(x)=ln(1+x^{1/n}+n^{-1/x}## with ##x>0## , ##n \in |N^+}## and ##f_{n2}(x)=\frac{x}{n}e^{-n(x+n)^2}## with ##x \in \mathbb{R} ## , ##n \in }|N^+}## The Attempt at a Solution 1) first series: ##f_{1n}## studying...

Hello MHB, I am pretty new with this serie I am supposed to find convergent or divergent. $$\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]$$ progress: $$\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]= \sum_{k=1}^\infty [\ln(\frac{k+1}{k})] = \sum_{k=1}^\infty[\ln(k+1)-\ln(k)]$$ so we got that...

Homework Statement Determine the radius of convergence of the given power serie . Ʃ(x^(2n))/n! n goes from 0 to infinity Homework Equations limit test ratio The Attempt at a Solution I am using the limit test ratio and I've got this : [n! * x^(2n+2)]/[(n+1)! * x^2n], then [n!*...

Homework Statement Consider the integrals \int_1^\infty \frac{k}{x^2+k^p\cos^2x}dm(x), where m is the Lebesgue measure. For what p do the integrands have an integrable majorant? For what p do the integrals tend to 0? Homework Equations The Attempt at a Solution Pick some...

Hello :) Could you tell me,why both of the Gauss-Seidel and Jacobi method,when we apply them at the tridiagonal matrix with the number 4 at the main diagonal and the number 1 at the first diagonal above the main and also the number 1 at the first diagonal under the main diagonal converge,but...

From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...

Homework Statement Find the radius of convergence and the interval of convergence Homework Equations A_n = Ʃ sum n =1 to infinity [((-1)^n) x^(2n+1)]/(2n+1)! The Attempt at a Solution All I thought was to use the ratio test so I did A_(n+1) /A_n = ((x^(2n+1))/(2n+1)!) (...

prove the convergence of the series and find the sum. please help me

Homework Statement An=[SIZE="5"]Ʃ(k)/[(n^2)+k] the sum is k=0 to n, the question is, to which value does the this series converge to Homework Equations i know for sure that this series converges, but could not figure out the value to whch it converges The Attempt at a Solution i...

Hi, I have a basic question about convergence. I have two sequences, x1, x2, ... and y1, y2, ..., where yn = f(xn) for some function f : ℝN → ℝ. I have shown that the sequence, y1, y2, ... converges. What conditions do I need on the function, f, to ensure that the sequence x1, x2...

Define $$f_{n}(x)=\frac{n^{1.5}x}{1+n^{2}x^2}$$ for x in [0,1]. Use Dominated convergence theorem to find the limit of the integral of f_n over [0,1]. I find that f_n converges to 0 so if I can find domination function I have shown integral is zero. Correct? I find f_n is dominated by function...

Let: gn(x) = 1 in [1/4 - 1/n2 to 1/4 + 1/ n2) for n = odd 1 in [3/4-1/n2 to 3/4 + 1/n2) for n = even 0 elsewhere Show the function converges in the L2 sense but not pointwise. My issue is in how I should use the definition of...

I have a question where I am supposed to show that a series does not converge uniformly, I get the majority of the question, but one part in the solution I can't see the rationale or how they decided on the result: It has to do with the partial sum: SN= (1 - (-x2)N+1)/ (1+x2) The...

If f_{n} \underset{n \to \infty}{\longrightarrow} f in L^{p}, 1 \leq p < \infty, g_{n} \underset{n \to \infty}{\longrightarrow} g pointwise and || g_{m} ||_{\infty} \leq M \forall n \in \mathbb{N} prove that: f_{n} g_{n} \underset{n \to \infty}{\longrightarrow} fg in L^{p} My attemp...

I need some examples of sequences some converges uniformly and some point wise Thanks in advanced

Homework Statement Where does the Taylor series converge? [You do not need to find the Taylor Series itself] f(x)=... I have a few of these, so I'm mainly curious about how to do this in general. The Attempt at a Solution I haven't really made an attempt yet. If I were to make an...

Homework Statement Use the Limit Comparison Test to determine if the series converges or diverges: Ʃ (4/(7+4n(ln^2(n))) from n=1 to ∞. (The denominator, for clarity, in words is: seven plus 4n times the natural log squared of n.) Homework Equations Limit Comparison Test: Let Σa(n) be the...

Let $$(T_{n}) $$be a sequence in $${B(l_2}$$ given by $$T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). $$Show that $$T_{n}->T$$ given by $$T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). $$ I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.

I have a problem: the radius of convergence for the power series \sum(-1)^n \frac{(x-3)^n}{(n+1)} is R=1. Determine the interval of convergence. What does this mean? can anyone help me solve this please

Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity. My approach is that lim a_n/b_n = 0 means that there exists N in natural number (N) for...

prove if ## a_{2k} \rightarrow l ## and ## a_{2k-1} \rightarrow l ## then ## a_n \rightarrow l ## where ## a_{2k} ## and ## a_{2k-1} ## are subsequences of ## a_n ## my attempt: since: ## a_{2k} \rightarrow l ## then ## \forall \epsilon > 0 ## ##\exists N_1 \in \mathbb{R}## s.t. ##2k > N_1...

Homework Statement Determine the values of x for which the following series converges. Remember to test the end points of the interval of convergence. ^{∞}_{n=0}\sum\frac{(1-)^{n+1}(x+4)^{n}}{n} Homework Equations I worked it down to |x+4|<1 ∴-5<x<-3 The Attempt at a Solution...

Homework Statement Is the sequence of function ##f_1, f_2,f_3,\ldots## on ##[0,1]## uniformly convergent if ##f_n(x) = \frac{x}{1+nx^2}##? 2. The attempt at a solution I got the following but I think I did it wrong. For ##f_n(x) = \frac{x}{1+nx^2}##, I got if ##f_n \to0## then we must...

Hi everyone, :) Can somebody give me a hint to solve this problem. :) Problem: Let \(f\) be a function defined on \([a,\,b]\) with continuous second order derivative. Let \(x_0\in (a,\,b)\) satisfy \(f(x_0)=0\) but \(f'(x_0)\neq 0\). Prove that, there is a neighbourhood of \(x_0\), say...

Homework Statement Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.2. The attempt at a solution Is this correct? Should I add more detail? Thanks for your help! Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to...