What is Convergence: Definition and 1000 Discussions

CONvergence is an annual multi-genre fan convention. This all-volunteer, fan-run convention is primarily for enthusiasts of Science Fiction and Fantasy in all media. Their motto is "where science fiction and reality meet". It is one of the most-attended conventions of its kind in North America, with approximately 6,000 paid members. The 2019 convention was held across four days at the Hyatt Regency Minneapolis in Minneapolis, Minnesota.

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  1. benorin

    Need some kind of convergence theorem for integrals taken over sequences of sets

    I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...
  2. I

    Convergence of alternating series

    Homework Statement Do the following series converge or diverge? ## \sum_{n=2}^\infty \frac{1}{\sqrt{n} +(-1)^nn}## and ##\sum_{n=2}^\infty \frac{1}{1+(-1)^n\sqrt{n}}##. Homework Equations Leibniz convergence criteria: If ##\{a_n\}_{k=1}^\infty## is positive, decreasing and ##a_n \to 0##, the...
  3. ognik

    MHB Please check this convergence test (#2)

    $ \sum_{n}\frac{1}{n.{n}^{\frac{1}{n}}} $ Now $\frac{1}{n}$ diverges and $\ne 0$ , so by limit comparison test: $ \lim_{{n}\to{\infty}} \frac{n.{n}^{\frac{1}{n}}}{n} = \lim_{{n}\to{\infty}} {n}^{\frac{1}{n}} = \lim_{{n}\to{\infty}} {n}^0 = 1$ (I think the 2nd last step may be dubious?)...
  4. ognik

    MHB Please check this convergence test

    $ \sum_{n} \ln\left({1+\frac{1}{n}}\right) $ $ \ln\left({1+\frac{1}{n}}\right) = \ln\left({1}\right) + \ln\left({\frac{1}{n}}\right) = 0 +\ln\left({{n}^{-1}}\right) = -\ln\left({n}\right)$ Now $\lim_{{n}\to{\infty}} -\ln\left({n}\right) \ne 0$, therefore the series diverges. (Also can you...
  5. Z

    MHB Series Convergence with Comparison Test

    Hey, I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!
  6. nomadreid

    Interval of convergence for Taylor series exp of 1/x^2

    Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...
  7. G

    MHB Radius of Convergence for $\sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$

    Radius of convergence of $\displaystyle \sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$. If I let $z^2 = x$ I get a series whose radius of convergence is $2$ (by the ratio test). How do I get from this that the original series has a radius of convergence equal to $\sqrt{2}$?
  8. L

    MHB Is the power series convergent?

    Hello! I have a problem with the following exercise, in which i must calculate the ray of a power serie. This is the power serie: \sum_{K=0}^{+\infty}(k+1)z^{k+1}. I decide to use the ratio test, and so i calculate \lim_{k\rightarrow +\infty}\frac{a_{n+1}}{a_{n}} for n going to infinity and i...
  9. N

    MHB Determine the convergence of the series

    Need help. Determine the convergence of the series: 1. sum (Sigma E) from n=1 to infinity of: 1/((2*n+3)*(ln(n+9))^2)) 2. sum (Sigma E) from n=1 to infinity of: arccos(1/(n^2+3)) I think the d'alembert is unlikely to help here.
  10. acdurbin953

    Find the interval of convergence

    Homework Statement Find the interval of convergence of the power series ∑(x-2)n / 3n Homework Equations ρn = |an+1| / |an| The Attempt at a Solution I got that ρn = | (x-2) / 3 |. I set my ρn ≤ 1, since this is when the series would be convergent. Manipulating that expression, I got that the...
  11. H

    MATLAB Matlab ode solvers - adding a separate convergence criteria

    Hi, is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped?
  12. I

    MHB Interval of Convergence for Power Series

    Hi hi, So I worked on this problem and I know I probably made a mistake somewhere towards the end so I was hoping one of you would catch it for me. Thank you! Pasteboard — Uploaded Image Pasteboard — Uploaded Image
  13. ognik

    MHB Convergence Confusion

    The Cauchy Ratio test says: If $ \lim_{{n}\to{\infty}}\frac{a_{n+1}}{a_n} < 1 $ then the series converges. OK. Now I read that for a power series (of functions of x), the same test also provides the interval of convergence, i.e. If the series converges, then $...
  14. kostoglotov

    Verifying the Fourier Series is in Hilbert Space

    The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...
  15. G

    Proving Radius of Convergence |z_0| = R for Power Series

    Homework Statement Given the power serie ##\sum_{n\ge 0} a_n z^n##, with radius of convergence ##R##, if there exists a complex number ##z_0## such that the the serie is semi-convergent at ##z_0##, show that ##R = |z_0|##. Homework EquationsThe Attempt at a Solution Firstly, since...
  16. D

    Convergence of Series: Finding x for Convergence | Homework Statement

    Homework Statement For which number x does the following series converge: http://puu.sh/lp50I/3de017ea9f.png Homework Equations abs(r) is less than 1 then it is convergent. r is what's inside the brackets to the power of n The Attempt at a Solution I did the question by using the stuff in...
  17. Z

    Proving convergence for integral

    Homework Statement Prove that for every a ∈ ℝ+ the following improper integrals are convergent and measure its value. ∫a∞exp(-at)dt Edited by mentor: ##\int_a^{\infty} e^{-at} dt## ∫1∞exp(-2at)dt Edited by mentor: ##\int_1^{\infty} e^{-2at} dt## The Attempt at a Solution For the first...
  18. M

    Very basic convergence lens question

    Hello. I have a very simple question that I need answered for my science project. I am doing a project on the effect of the convexity of the lens on the intensity of converged light. (Lux?) I am using a class set which I haven't been able to get my hands on yet, but we are expected to be...
  19. S

    Interval of convergence of a linear operator

    Homework Statement A function of a hermitian operator H can be written as f(H)=Σ (H)n with n=0 to n=∞. When is (1-H)-1 defined? Homework Equations (1-x)-1 = Σ(-x)n= 1-x+x2-x3+... The Attempt at a Solution (1-H)-1 converges if each element of H converges in this series, that is (1-hi)-1...
  20. T

    Simple 4 bus Newton-Raphson matlab not converging

    Homework Statement All the necessary data is in the code, I'm just trying to converge NR, I decided to use the equation S = V^2 / Z since I had the admittance matrix and powers (needed voltages) I think my simple algorithm has a slight issue I can't find. Homework Equations Thank you! The...
  21. M

    Alternating series test for convergence

    Homework Statement Homework Equations The Attempt at a Solution I don't get how they got what's stated in the above picture. Where does 1/2 and n/(n + 1) come from? Can't you just show that an + 1 ≤ an?
  22. B

    Finding the rate of convergence for a markov chain

    Homework Statement For the following Markov chain, find the rate of convergence to the stationary distribution: \begin{bmatrix} 0.4 & 0.6 \\ 1 & 0 \end{bmatrix} Homework Equations none The Attempt at a Solution I found the eigenvalues which were \lambda_1=-.6 or \lambda_2=1 . The...
  23. W

    Marginal PMG of of 2 random variables with Joint PMF

    Homework Statement Consider two random variables X and Y with joint PMF given by: PXY(k,L) = 1/(2k+l), for k,l = 1,2,3,... A) Show that X and Y are independent and find the marginal PMFs of X and Y B) Find P(X2 + Y2 ≤ 10) Homework Equations P(A)∩P(B)/P(B) = P(A|B) P(A|B) = P(A) if independent...
  24. O

    MHB Triangle Inequality and Convergence of ${y}_{n}$

    Let ${y}_{n}$ be a arbitrary sequence in X metric space and ${y}_{m+1}$ convergent to ${x}^{*}$ in X...İn this case by using triangle inequality can we say that ${y}_{n}\to {x}^{*}$
  25. I

    Convergence of trigonometric(Fourier) series

    Homework Statement Which of the following series is point-wise convergent, absolutely convergent? Which ones are ##L^2(-\pi,\pi)##-convergent. A) ##\sum_1^\infty \frac{\cos n \theta}{n+1}## B) ##\sum_1^\infty \frac{(-1)^n\cos n \theta}{n+1}## Homework Equations Abel's test:[/B] Suppose ##\sum...
  26. I

    Understanding convergence in norm, uniform convergence

    Homework Statement Find an example of a sequence ##\{ f_n \}## in ##L^2(0,\infty)## such that ##f_n\to 0 ## uniformly but ##f_n \nrightarrow 0## in norm. Homework Equations As I understand it we have norm convergence if ##||f_n-f|| \to 0## as ##n\to \infty## and uniform convergence if there...
  27. S

    Does Convergence in the Mean Imply Ordinary Convergence?

    Homework Statement 1. Consider the sequence $$\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5},\frac{1}{6}, \ldots$$ For which values ##z \in \mathbb{R}## is there a subsequence converging to ##z##? 2. Prove that...
  28. SteliosVas

    Indefinite integral and proving convergence

    Homework Statement okay so the equation goes: ∫(x*sin2(x))/(x3-1) over the terminals: b= ∞ and a = 2 Homework Equations Various rules applying to the convergence or divergence of integrals such as the p-test, ratio test, squeeze test etc The Attempt at a Solution Okay so I have tried...
  29. B

    Convergence of a sum over primes

    I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers, then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges. I have tried...
  30. R

    Power series where radius of convergence > lower limit

    Homework Statement Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha = \limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...
  31. A

    Convergence and Cauchy Criterion

    Homework Statement Suppose the sequence (xn) satisfies |xn + 1 - xn| < 1/n2, prove that (xn) is convergent. Homework Equations |xn - xm| < ɛ The Attempt at a Solution If m > n, then |xn - xm| < |xn - xn + 1| + |xn + 1 - xn + 2| + ... + |xm - 1 - xm| < 1/n2 + 1/(n+1)2 + ... + 1/(m - 1)2 <...
  32. A

    Proving Convergence of a Sequence Using Cauchy Criterion

    Homework Statement Show that the sequence {xn}: xn := (21/1 - 1)2 + (21/2 - 1)2 + ... + (21/n - 1)2 is convergent. Homework EquationsThe Attempt at a Solution If n > m, |xn - xm| = (21/n - 1)2 + (21/(n-1) - 1)2 + ... + (21/(m+1) - 1)2 < (21/n)2 + (21/(n-1))2 + ... + (21/(m+1))2 < (21/(m+1))2 +...
  33. M

    MHB Convergence of a geometric series

    Hi everyone, I am generally familiar with convergent series. However, in one economics paper (Becker&Tomes 1979), I found the following that confuses me:$$\sum_{j=0}^{k} \beta^{j} h^{k-j} = \beta^{k}(k+1)\quad \text{if} \quad\beta =h$$ however, $$\sum_{j=0}^{k} \beta^{j} j^{k-j} =...
  34. P

    Advection equation stability for explicit scheme

    Hello, I have a typical 1D advection problem where a cold fluid flows over a flat plate. I did an energy balance to include conduction, convection and friction loss and I got the PDE's for the fluid and the solid. I used finite differences to solve the system as T(x, t) for both fluid and...
  35. R

    Calculus 2 interval of convergence -- checking end points

    Homework Statement...
  36. F

    MHB What is the Condition for Convergence of Sequence $p_{n}$?

    I literally don't know how to solve this one. I hope you can help me. :)
  37. C

    Convergence of a hypergeometric

    The hypergeometric function, ##{}_{2}F_1(a,b,c;z)## can be written in terms of a power series in ##z## as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}\,\,\,\,\,\text{provided}\,\,\,\,|z|<1$$ So we may reexpress any hypergeometric function as a...
  38. R

    Finding the radius of convergence of a power series

    Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...
  39. ognik

    MHB Range of Uniform convergence of dirchlet series

    Find the Range of Uniform convergence of $ \zeta\left(x\right) = \sum_{n=1}^{\infty}\frac{1}{{n}^{x}} $ Using the Weierstrass-M test, I get this converges for $ 1 \lt x \lt \infty $ But the book's answer is $ 1 \lt s \le x \lt \infty $? I have scoured the book but can't see why they say it...
  40. S

    Telescoping series convergence question

    Homework Statement [/B] Hello, this problem is from a well-known calc text: Σ(n=1 to ∞) 8/(n(n+2)Homework Equations [/B] What I have here is decomposingg the problem into Σ(n=1 to ∞)(8/n -(8/n+2)The Attempt at a Solution I have the series sum as equaling (8/1-8/3) + (8/2-8/4) + (8/3-8/5) +...
  41. P

    Questions on Newton's Method: Investigating Convergence & Initial Guesses

    I have three questions regarding Newton's method. https://en.m.wikipedia.org/wiki/Newton-Raphson#Failure_of_the_method_to_converge_to_the_root According to this wikipedia article, "if the first derivative is not well behaved in the neighborhood of a particular root, the method may overshoot, and...
  42. ognik

    MHB How to Apply Gauss's Convergence Test to a Hypergeometric Series?

    I found the interval of convergence for a hypergeometric series as |x| < 1, now I believe that I need to apply 'Gauss's test' to check the end point(s). For $ \left| x \right|=1 $ my $ \left| \frac{{a}_{n}}{{a}_{n+1}} \right| = \left|...
  43. J

    Doubt about convergence test on differential equations

    I will try to explain this with an analogy. Let's have this equation: x^2 =9 And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put...
  44. L

    Convergence of improper integrals

    What is the difference between \int_{-\infty}^{\infty} \frac{x}{1+x^2}dx and \lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ? And why does the first expression diverge, whilst the second converges and is equal to zero?
  45. J

    Proof Taylor series of (1-x)^(-1/2) converges to function

    Hello, I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this: Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n My attempt is to use the lagrange error bound, which is...
  46. B

    MHB Problem evaluating the limit to find the radius of convergence

    Hi everyone, I am trying to evaluate the radius of convergence for the following power series: (k!(x-1)k)/((2k)(kk)) I have begun by trying to compute L = lim k-->inf (an+1/an). To then be able to say R = 1/L. So far i have L = lim k--> inf (kk(k+1)!)/(2(k+1)k+1k!) From here i am having...
  47. B

    MHB Radius of Convergence: Evaluate & Ignoring Extra Vars

    I am attempting to evaluate the radius of convergence for a series that goes from k=0 to infinity. The series is given by (k*x^k)/(3^k). I have begun by using the ratio test and have gotten to the point L = (k+1)*x/3k Now i know i can find out the radius of convergence by simply saying R =...
  48. evinda

    MHB Understanding the Radius of Convergence of e^x and its Series Expansion

    Hello! (Wave) $$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$ i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$. Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$? Do...
  49. Coffee_

    Convergence of the sequence from Heron's method.

    ##a_1=1## ##a_{n+1}=\frac{1}{2} ( a_{n} + \frac{b}{a_n} )## This should converge to ##\sqrt{b}## but I seem not to be able to prove this. Could someone give me a hint.
  50. Draconifors

    Convergence of a Complex Series

    Homework Statement "Determine whether the following series converge or diverge. If the series is geometric or telescoping, find its sum.": ## \left ( \sum_{k=1}^\infty2^{3k} *3^{1-2k} \right)## Homework Equations [/B] The different tests for convergence? The Attempt at a Solution Ok...
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