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. STEMucator

    Convergence of Sequences in [0,1]

    Homework Statement Determine the convergence, both pointwise and uniform on [0,1] for the following sequences : (i) ##s_n(x) = n^2x^2(1 - cos(\frac{1}{nx})), x≠0; s_n(0) = 0## (ii) ##s_n(x) = \frac{nx}{x+n}## (iii) ##s_n(x) = nsin(\frac{x}{n})## Homework Equations ##s_n(x) →...
  2. W

    Convergence: Test for Convergence of ∫ sin∅/sqrt(pi-∅) d∅

    Homework Statement Test for convergence or divergence. Homework Equations \displaystyle\int_0^∏ {\frac{sin∅}{\sqrt{pi-∅}} d∅} The Attempt at a Solution the solution manual does the following: pi -∅ = x \displaystyle\int_0^pi {\frac{sinx}{\sqrt{x}} d∅} 0 <=...
  3. D

    Is the Sequence {a_n} Convergent?

    Homework Statement Let {a_n} be a sequence | (a_n+1)^2 < (a_n)^2, 0 < (a_n+1) + (a_n). Show that the sequence is convergent Homework Equations n/a The Attempt at a Solution So I am feeling like monotone convergence theorem is the way to go there. It seems to me that (a_n+1)^2 <...
  4. stripes

    Uniform convergence for heat kernel on unit circle

    Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...
  5. P

    Series convergence vs. divergence

    Simple question: Are there any series which we don't know whether or not they converge?
  6. C

    Convergence in distribution (weak convergence)

    Homework Statement Consider the function f_n(x)=n\cdot I\left[|x|<\frac{1}{2n}\right] considered as a distribution in D'(\mathbb{R}), where I denotes the indicator function. Recall that f_n converges to \delta_0, the delta distribution, in D'(\mathbb{R}). Show that f_n^2-n\delta_0...
  7. STEMucator

    Series Converg. Hmwk: Determine Convergence of \sum(-1)^n n/(n^p + (-1)^n)

    Homework Statement Really tough series to work with. Determine the convergence ( absolute or conditional ) or divergence of : ##\sum_{n=2}^{∞} \frac{(-1)^n n}{n^p + (-1)^n}## Homework Equations ?? Series tests? The Attempt at a Solution This series is really ugly. I'm not sure how to...
  8. M

    EM algorithm convergence KF log likelihood decrease

    Hi everyone, Im running the KF to learn parameters of a model, the log likelihood of the p(Y_{k}|Y_{k-1}), however decreases. Can anyone advise, does this mean my implementation is wrong or can this just be the case. Advice appreciated Thanks
  9. MarkFL

    MHB Convergence of numeric schemes

    A while back, I dug out a topic I worked on many years ago after taking a course in ordinary differential equations and I was left with an unanswered question, which I thought I would post here. While my question arose from studying numeric methods for approximating the solutions to ODEs, I feel...
  10. G

    Convergence of Mean in Probability - How to Prove it?

    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...
  11. C

    MHB Confirm Answers on Homework Sheet: Subsequence Convergence

    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 examples for those that...
  12. D

    Bounded sets, Limits superior and convergence

    (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...
  13. T

    The Continutiy and the Convergence.

    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...
  14. C

    Mesh Convergence Issue In Ansys

    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...
  15. L

    Limit of integral lead to proof of convergence to dirac delta

    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...
  16. alyafey22

    MHB Proof the convergence of a gamma sum

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

    Convergence in Probability am I doing something wrong?

    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...
  18. P

    Determining convergence of a sum

    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...
  19. stripes

    Good kernels, convergence, and more

    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...
  20. S

    So, the infinite series converges for a>2 and diverges for a=2.

    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...
  21. T

    Exact solutions and the convergence of eulers method

    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...
  22. S

    How do i evaluate the convergence

    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%?
  23. A

    Cdf of a discrete random variable and convergence of distributions

    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...
  24. A

    Therefore, since P(A) = 0, we have convergence in probability.

    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) =...
  25. stripes

    Fourier series coefficients and convergence

    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...
  26. T

    Convergence problem (nth-term test)

    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)) /...
  27. L

    MHB Series convergence with a floor

    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...
  28. B

    Radius of convergence and 2^1/2

    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...
  29. C

    QED perturbation series convergence versus exact solutions

    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...
  30. L

    MHB Help with Monotonic Sequence Convergence

    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.
  31. L

    MHB Solving Series Convergence Problems: 1+ \frac{1}{3}-\frac{1}{2}+\frac{1}{5}+...

    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}{...
  32. P

    MHB Exponent of convergence of a sequence of complex numbers

    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 $\{z_j\}$...
  33. K

    MHB Problem concerning martingale convergence theorem

    My goal is providing a proof based on martingale convergence theorems for the following fact: Series $S_n:=\sum\limits_{k=1}^n X_k$ of independenet random variables converges in distribution. Prove that $S_n$ converges almost certainly. I suppose these are not sufficent assuptions about $X_n$...
  34. F

    Radius of Convergence: Complex Series Need Not Be Defined Everywhere

    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...
  35. P

    MHB Families of holomorphic functions and uniform convergence on compact sets

    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...
  36. P

    Determining the radius of convergence

    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...
  37. E

    Convergence of non increasing sequence of random number

    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
  38. J

    Uniform Convergence and the Uniform Metric

    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...
  39. B

    Proof for a Sequence Convergence

    \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...
  40. F

    Prove the convergence of a limit

    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...
  41. A

    A question about uniform convergence

    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...
  42. M

    Convergence of a functional series (analysis)

    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...
  43. phosgene

    Convergence of \sum{\frac{2^n}{3^n - 1}} using the limit comparison test

    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...
  44. M

    Exploring the Convergence of \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)}

    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...
  45. F

    Convergence of a sequence + parametre

    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...
  46. C

    About interesting convergence of Riemann Zeta Function

    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...
  47. A

    Radius of convergence (power series) problem

    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
  48. F

    Convergence of a Sequence: Proving Existence of Limit Using Cauchy Sequences

    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...
  49. H

    Radius and Interval of Convergence

    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...
  50. M

    Analytic Functions and Intervals of Convergence

    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...
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