Convergence Definition and 1000 Threads

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  1. H

    Ratio test for finding radius of convergence

    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...
  2. A

    How Do You Choose Comparison Limits in Series Convergence Tests?

    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...
  3. A

    Proving Series Convergence: \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}

    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!
  4. M

    MHB Series Convergence: Showing Convergence & Sum Equivalence

    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...
  5. QuantumCurt

    Determine the convergence or divergence of the infinite series

    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}...
  6. S

    MHB More Convergence & Divergence with sequences

    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?
  7. S

    MHB Tricky question Considering Divergence and Convergence

    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 3n^2 = \infty...
  8. S

    MHB Convergence and Divergence with Series

    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 understand why this is...
  9. A

    Prove Convergence of Sequence Defined by f(an) in R

    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...
  10. D

    Radius and interval of convergence

    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
  11. M

    Question on fourier series convergence

    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!
  12. A

    Proving convergence given inequalities of powers

    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...
  13. B

    Uniform Convergence of a Sequence of Functions

    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}...
  14. kq6up

    Test Convergence: Sum of i^n/n | Chris Maness

    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...
  15. Darth Frodo

    Proving convergence of recursive sequence.

    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.
  16. D

    MHB Region of Convergence and Inverse Laplace

    How do I find all the possible ROC for a transfer function written as \[ H(s) = \frac{(s - 2)^{n_1}}{(s + 2)^{n_2}(s + 1)^{n_3}(s - 1)^{n_4}} \] where \(n_i\in\mathbb{N}\).
  17. D

    MHB Region of Convergence: Re${s} > -3 for $\beta$

    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...
  18. M

    Explaining Klauder's "Modern Approach" Convergence Question

    From Klauder's "Modern Approach to Functional Integration"... The integrals go over all of ℝ. What confuses me most is that this is pretty much at the beginning of the book and the author ocassionally explains rather obvious things but every now and then: something like this. Can anyone...
  19. G

    Convergence of 10^-2^n. Linear, quadratic, cubic, quartic, hectic

    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...
  20. S

    MHB Is the Integral \int^0_{-\infty} \frac{1}{3 - 4x} dx Divergent or Convergent?

    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...
  21. Krizalid1

    MHB Convergence of Series: $\sum a_nb_n$

    Let $a_n$ and $b_n$ be sequences in $\mathbb R.$ Show that if $\displaystyle\sum b_n$ converges and $\displaystyle\sum|a_n-a_{n+1}|<\infty,$ then $\displaystyle\sum a_nb_n$ converges.
  22. S

    Convergence Proof: Show {n2 - n + 5} Increasing & {xn} Convergent

    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...
  23. S

    Proof uniform convergence -> continuity: Why use hyperhyperreals?

    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...
  24. vanceEE

    Convergence by Comparison Test

    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...
  25. L

    Range of uniform convergence for a series

    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...
  26. Z

    Convergence of a Sequence: How to Determine and Find the Limit?

    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...
  27. S

    MHB Pointwise convergence implies uniform 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...
  28. F

    MHB Proving Absolute Convergence of a Real-Valued Function on a Sigma Algebra

    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...
  29. F

    MHB Pointwise Conv. | Does $f_{n}$-$f$ -> 0 for Each x?

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

    How to Find the Interval of Convergence for this Series?

    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
  31. E

    Convergence in probability distribution

    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...
  32. C

    Why are the stresses not converging in my Ansys plane stress model?

    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...
  33. F

    Study of the convergence (pointwis&uniform) of two series of functions

    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...
  34. Petrus

    MHB Finding Convergence of Series: $\sum_{k=1}^\infty[\ln(1+\frac{1}{k})]$

    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...
  35. B

    Determining the raidus of convergence

    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!*...
  36. J

    Solving Convergence Problem: Integrals with Lebesgue Measure

    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...
  37. evinda

    MHB Are Gauss-Seidel and Jacobi Methods Guaranteed to Converge on Certain Matrices?

    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...
  38. N

    Do Fourier transforms always converge to 0 at the extreme ends?

    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...
  39. J

    Finding Radius & Interval of Convergence

    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)!) (...
  40. A

    Convergence of Series: Proving and Finding the Sum

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

    Can Convergence Theories Be Categorized as Countable and Uncountable?

    I'm trying to organize my thought processes about real analysis, using general questions to motivate the theory, in the hopes of using this format for when I study functional analysis or something, so it doesn't feel like 50 new ideas & instead is the modification of previously existing ideas in...
  42. O

    Series Convergence: An=Ʃ(k)/[(n^2)+k] - Find Value

    Homework Statement An=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 did the...
  43. D

    What Conditions Ensure Convergence of Sequences?

    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...
  44. S

    A strong version of Dominated Convergence Theorem - Real Analysis

    Homework Statement Let (g_{n})_{n \in \mathbb{N}} a sequence functions integrable over \mathbb{R}^{p} such that: g_{n} (x) \longrightarrow g(x) almost everywhere in \mathbb{R}^{p}, where g is a function integrable over \mathbb{R}^{p}. Given (f_{n})_{n \in \mathbb{N}} a sequence of...
  45. F

    MHB Can the Dominated Convergence Theorem Apply to f_n(x) on [0,1]?

    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 g...
  46. T

    Convergence in Uniform and L2 sense, function interpretation

    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...
  47. T

    Proving Non-Uniform Convergence: Understanding the Role of Singular Points

    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...
  48. S

    Why is the last step of my proof for convergence in L^{p} space correct?

    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...
  49. A

    MHB Examples of Uniformly, point wise convergence

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

    Where Can I Find Information on Testing the Convergence of Taylor Series?

    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...
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