Convergence Definition and 1000 Threads

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. K

    Weak convergence of orthonormal sequences in Hilbert space

    So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such. I've come to understand that this property follows...
  2. Fredrik

    Convergence in Measure: Understanding and Proving Almost Everywhere Convergence

    Not sure where to post about measure theory. None of the forums seems quite right. Suppose that ##(X,\Sigma,\mu)## is a measure space. A sequence ##\langle f_n\rangle_{n=1}^\infty## of almost everywhere real-valued measurable functions on X is said to converge in measure to a measurable...
  3. J

    Convergence of several improper integrals

    There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...
  4. B

    Convergence of C[0,2*pi] with f(x)=sin(x) and sup|f(x)|=1

    Folks, For C[0,2*pi] and given a function f(x)=sin(x) the supremum |f(x)|=max|f(x)| for x in [a,b] I calculate the sup|f(x)| to be = 0 but my notes say 1. The latter answer would be the case if f(x) was cos(x)...right?
  5. T

    Determining convergence or divergence

    Homework Statement Use a valid convergence test to see if the sum converges. Ʃ(n^(2))/(n^(2)+1) Homework Equations Well, according to p-series, I'd assume this sum diverges, but I don't know which test to use. The Attempt at a Solution I probably can't do this, but I was think...
  6. H

    How to increase the convergence speed?

    Hi, I'm using the finite difference method to solve both the Reynolds Equation and the Poisson equation in order to calculate the air pressure of a porous air bearing. The Reynolds equation describes the flow in the channel between the rotor and stator, while the Poisson equation the flow...
  7. T

    Divergence and convergence question

    Is the sum of 2 divergent series Ʃ(an±bn) divergent? From what I have learned is that it is not always divergent. Is this true? I believe that is what the picture i included is saying, but i maybe miss interpreting it. Also, is the product of 2 divergent series divergent or convergent?
  8. M

    Prove cauchy sequence and thus convergence

    Let (Xn) be a sequence satisfying |Xn+1-Xn| ≤ λ^n r Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.
  9. V

    Weak convergence of the sum of dependent variables, question

    Hi guys, Problem: Let {Xn},{Yn} - real-valued random variables. {Xn}-->{X} - weakly; {Yn}-->{Y} weakly. Assume that Xn and Yn - independent for all n and that X and Y - are independent. Fact that {Xn+Yn}-->{X+Y} weakly, can be shown using characteristic functions and Levy's theorem...
  10. T

    Difference between convergence of partial sum and series

    I am a little confused as to notation for convergence. I included a picture too. If you take a look it says "then the series Ʃan is divergent" Does the "Ʃan" just mean the convergence as to the sum of the series, or the lim an as n→ ∞ nth term? I believe it is the sum of the series but I...
  11. T

    Finding the radius of convergence

    Finding the radius of convergence... Homework Statement 1+2x+(4x^(2)/2!)+(8x^(3)/3!)+(16x^(4)/4!)+(32x^(5)/5!)+... Homework Equations I would use the ratio test. Which is... lim as n→∞ (An+1/An) The Attempt at a Solution I know what to do to find the answer, but I don't know...
  12. T

    Using ratio test to find radius of convergence

    Homework Statement Ʃ((x-3)^(n)) / (n*2^(n)) Homework Equations lim as n→ ∞ (An+1 / An) The Attempt at a Solution When dividing two fractions, invert the second and multiple to get what you see below. (x-3)^(n+1)/((n+1)*2^(n+1)) * (n*2^(n))/((x-3)^(n)) Do some cross...
  13. J

    Question about radius of convergence of fractional power series

    Suppose I have the Laurent series with region of convergence given below: f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5} Can I conclude the Laurent-Puiseux series: f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n has a region of convergence...
  14. H

    I just dont get this proof of convergence of even and uneven sequence coeffi. ?

    Homework Statement Show that if Lim(n-->inf.)(a_(2n)-->L) and Lim(n-->inf.)(a_(2n+1)-->L) then Lim(n-->inf.)(a_n-->L). The Attempt at a Solution I just don't get this; I can see the big picture though. If the odd coeffictions of a sequences goes towards one the same number as the even...
  15. 1

    I want to know more about series convergence (elementary)

    I am able to use a variety of methods to check to see if a series converges, and I can do it well. However, it's not something I feel like I've intuitively conquered. I don't understand why the series 1/x diverges. I mean, I do, in that I know the integral test will give me the limit as x ->...
  16. T

    Difficult series convergence proof

    Homework Statement Show that given some ε > 0, there exists a natural number M such that for all n ≥ M, (a^n)/n! < ε Homework Equations The Attempt at a Solution Ok so I know this seems similar to a Cauchy sequence problem but its not quite the same. So I am looking for a...
  17. Y

    Pointwise convergence of (x^n)/(1 + x^n)

    Homework Statement Find the function that (x^n)/(1 + x^n) converges to as n goes to infinity, on the interval [0,2] Homework Equations The Attempt at a Solution I've worked out the fact that on the interval [0,1) it converges to 0, and when x is 1 it converges to 1/2, but for the...
  18. R

    Is the following correct? (concerns sets and convergence)

    Let A = {(x,y) in R^2 | x^2 + y^2 <= 81} Let B = {((x,y) in R^2 | (x-10)^2 + (y-10)^2 <= 1} then here "A intersection B" is the empty set. Then let x_n be the sequence (0,10-(2/n)) which is a sequence in A and y_n be the sequence (10/n,10) which is a sequence in B. would |x_n - y_n| tend...
  19. S

    Complex Series, Region of Convergence

    Good evening, I'm an electrical engineering student questioning my answer to this series Region of Convergence problem. Ʃ(0,inf) (n(n-1)(z+5i)^n)/n Using the ratio test lim n-> |an+1/an| I was able to get it down to lim n->|n(z+5i)/(n-1)| which gave |inf/inf| = 1, which means the test...
  20. P

    Radius of convergence: 1/(1+x^2) about 1, using only real analysis

    I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...
  21. F

    Nested radicals and its convergence

    Homework Statement This is supposed to be really easy, but I don't think my answer is good Consider this \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} I was hinted that a_{n + 1} = \sqrt{1 + a_n} for all n ≥ 0 and I am supposed to show that the sequence convergees The Attempt at a...
  22. A

    Finding the divergence or convergence of a series

    Ʃ ,n=1,∞, (2/n^2+n) Does this series converge or diverge? Im not sure how to start can i use the comparison test here?
  23. L

    Series Convergence and Divergence

    Homework Statement Determine if the following series converges or diverges. If it converges determine its sum. Ʃ1/(i2-1) where the upper limit is n and the index i=2 Homework Equations The General Formula for the partial sum was given: Sn=Ʃ1/(i2-1)=3/4-1/(2n)-1/(2(n+1) The...
  24. F

    Determining convergence of (-1)^n/ln(n) as n goes to infinity

    Homework Statement Determine whether (-1)^n/ln(n!) is divergent, conditionally convergent, or absolutely convergent. Homework Equations None, really? :SThe Attempt at a Solution Okay, so I know the series converges by the Alternating Series test - terms are positive, decreasing, going to...
  25. T

    Convergence of a sequence in a metric space

    Homework Statement For x,y \in\mathbb{R} define a metric on \mathbb{R} by d_2(x,y) = |\tan^{-1}(x) - \tan^{-1}(y) | where \tan^{-1} is the principal branch of the inverse tangent, i.e. \tan^{-1} : \mathbb{R} \to (-\pi/2 ,\pi/2). If (x_n)_{n\in\mathbb{N}} is a sequence in \mathbb{R} and...
  26. H

    What Is the Interval of Convergence for the Series Summation?

    Homework Statement Find the interval of convergence of each of the following Ʃ^{∞}_{n=0} (\frac{3^{n}-2^{2}}{2^{2n}}(x-1)^{n}) Homework Equations Please refer to attachment The Attempt at a Solution Please refer to attachment. All I want to know is that I'm doing this problem...
  27. D

    Conditional and absolute convergence (of series)

    I was reading this article of wikipedia: Conditional and absolute convergence It says: "An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long." Is that a characterization of...
  28. H

    Is the Series ∑ (sin(1/n)/√n) Convergent?

    Homework Statement Ʃ^{∞}_{n=1} \frac{sin(1/n}{\sqrt{n}} Homework Equations The Attempt at a Solution lim_{n→∞} \frac{sin(1/n}{\sqrt{n}}= lim_{n→∞} \frac{-2cos(1/n)}{n^{3/2}} with l'hospital rule = 0 since lim_{n→∞}=0 therefore the series is convergent Do you think I did this...
  29. T

    Pointwise Convergence in Metric Space (C[a,b],d_{\infty}) | Homework Solution

    Homework Statement Suppose a sequence (f_n)_{n\in\mathbb{N}} converges to a limit f in the metric space (C[a,b],d_{\infty}) (continuous real valued functions on the interval [a,b] with the uniform metric.) Show that f_n also converges pointwise to f; that is for each t\in [a,b] we have...
  30. M

    Convergence Proof (As Part of Geometric Series Sum)

    Homework Statement I am trying to prove the sum of a geometric series, but one of the steps involves deriving this result: \lim_{n\to\infty}r^{n}=0 so that you can simplify the sum of a geometric series, where I have got to this stage: S_{\infty} = \frac{a(1-r^{\infty})}{1-r}...
  31. P

    Analysis: Absolute convergence, rearranging terms, power series question

    Homework Statement I can't figure out what my professor means in the last two lines: I'm trying to prove why the product of two analytic functions is analytic, and I think that I am going to need to use a similar construction. The Attempt at a Solution So far to prove the product...
  32. P

    Tricky series radius of convergence question (analysis course)

    Homework Statement Find the radius of convergence of sum from 1 to n of 1/(n^n) * x^(2^n) Homework Equations The Attempt at a Solution Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent
  33. M

    Comparison test to determine convergence

    Use a comparison test to determine whether the series \sum (n+1)/(n^{2}+n+1) diverges or converges. I started out by simplifying the series to 1/n+1 and then from there I compared it to 1/n, which converges. 1/n is greater than 1/n+1 so based on the comparison test, the original series...
  34. J

    Convergence: Root Test Inconclusive

    Homework Statement I'm this series to see if it's convergent or divergent. I tried the root test, but it came out inclusive, and now I am trying to figure out if the ration test works. The only thing I'm asking which would be the right test for this. Homework Equations ∞Ʃn=1...
  35. H

    Proof on Convergence of Sequence Given Info on Odd/Even Subsequences

    Homework Statement Given that limit of s_{2n} is L and limit of s_{2n+1} is L, prove that lim s_{n} is also L. Homework EquationsThe Attempt at a Solution This seems very obvious: If the even terms of a sequence approach a number and the odd terms of that sequence approach the same number...
  36. T

    Complex Power Series Convergence Help

    Homework Statement I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence. Homework Equations As an example: Σ(z+3)k2 with...
  37. M

    Planewave cutoff energy convergence

    Hello everyone, I have some problems with the convergence for the planewave cutoff energy, I tried to find the correct PWcE for a system with 2 atoms, I've determined the minimum k-point mesh with success, but when I compute the Total energy varying the PWcE, it seems that the convergence...
  38. TheFerruccio

    Extremely simple question regarding the convergence of a series

    First off, this is not a homework question. I am helping someone with an alternating series, and, for some reason, I am finding myself completely baffled by this one. I have an alternating series that takes the form: \sum\limits_{j=1}^\infty(-1)^j\frac{\sqrt{j}}{j+5} I know that the...
  39. M

    How Can I Prove Uniform Convergence of This Function as ρ→0?

    Probably is a silly question, but how could I prove that the function (expressed in polar coordinates) \left(\rho^4\cos^2{\theta} + \sin^3{\theta}\right)^{\frac{1}{3}} - \sin{\theta} converges to 0 as rho->0 uniformely in theta (if it is true, of course)?
  40. Z

    Convergence of sqrt(2+sqrt(sn)) = s_n+1

    Homework Statement Show convergence of s_{n+1}= \sqrt{2+\sqrt{s_n}} where s_1 = \sqrt{2} and that s_n<2 for all n=1,2,3... Homework Equations Let {p_n}be a sequence in metrice space X. {p_n} converges to p iff every neighborhood of p contains p_n for all but a finite number of n.The...
  41. S

    Need help proving convergence of this series

    n! / nn I have proven that the sequence converges numerically, but I can't do it analytically, and can't do anything for the series (maybe it the series doesn't converge?)
  42. L

    Convergence of Series: Dyadic Criterion for Series Involving Logs

    Homework Statement "Determine whether the following series converge: \sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}} and \sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}} Homework Equations The convergence/divergence tests (EXCEPT INTEGRAL TEST): Ratio Dyadic Comparison P-test...
  43. nomadreid

    Radius of convergence around a=2 for ln(x)

    When I use the calculation from Wikipedia that says that the radius of convergence of a series is lim as n goes to infinity of |an/an+1|, I get for the Taylor series expansion of ln(x) around a=2 the answer of an infinite radius of convergence, which would mean that it would be valid everywhere...
  44. G

    Does Convergence of (sn) and (sntn) Imply (tn) Converges?

    Hello was wondering if anyone could help me prove that: Suppose (sn) converges to s not equal to 0 and ( sntn) converges to L. Prove that (tn) converges
  45. T

    Alternating Series Tests: Understanding Conditional & Absolute Convergence

    I have a question about the ratio test. Suppose it proves inconclusive, we must than use another test to check for conditional convergence - 1) this test has to be associated with an alternating series, such as the Alternating Series Test, correct? (we wouldn't be able to use something like...
  46. T

    Convergence on [-M,M] for any M implies convergence on R?

    Good day dear fellows. I am given the following series h(x) = \sum_{n=1}^{\infty} \frac{1}{x^2+n^2}. It is requested that I show that h(x) is continuous on R. I did the following: use the Weirerstrass M-test to show uniform convergence, and then, using the continuity of the functions...
  47. K

    Radius & interval of convergence of power series

    Homework Statement i was doing this exercise and came across this example. ∞ Ʃ (x^n)/ln(n+1) n=1 The Attempt at a Solution i know you have to do the ratio test which is lim | a(n+1)/a(n)| n>∞ i got to lim | [x ln(n+1)] /ln(n+2) | n>∞ and have no idea how to continue? is...
  48. I

    Radius of convergence (Power series)

    Homework Statement Hi there, I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.Take for example: The...
  49. T

    Interval of Convergence: Is it an Interval of Convergence Question?

    Homework Statement I've attached the question Homework Equations The Attempt at a Solution I'm not exactly sure how to do this question. Is it an interval of convergence question where i simply let log(1+2x) < 1 and solve for x??
  50. D

    How can i prove these two convergence theorem?

    1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent 2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ..., then {an/bn} is also convergen
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