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. K

    Limit of a Series with Unknown Variable K

    Homework Statement Determine whether the following are convergent, divergent or oscillating. Homework Equations Please see the attachment The Attempt at a Solution Please see the attachment. I am unsure about this as when I plot a graph without K its convergence
  2. A

    Convergence or divergence

    The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction? Thanks
  3. phosgene

    Uniform convergence of sequence of functions

    Homework Statement Let f_{n}(x)=\frac{x}{1+x^n} for x \in [0,∞) and n \in N. Find the pointwise limit f of this sequence on the given interval and show that (f_{n}) does not uniformly converge to f on the given interval. Homework Equations The Attempt at a Solution I found that the pointwise...
  4. P

    Negation of definition of convergence

    The definition of convergence is given by : ## \forall \epsilon > 0, \exists N \in \mathbb{R} ## such that ## |x_n - l | < \epsilon ## ## \forall n \in \mathbb{N} ## with ## n > N ## negate this statement and prove that the sequence ## x_n = (-1)^nn ## is divergent using only the negation of...
  5. E

    Uniform convergence of a product of functions

    Homework Statement Let \left[a,b\right] be a closed bounded interval, f : [a,b] \rightarrow \textbf{R} be bounded, and let g : [a,b] \rightarrow \textbf{R} be continuous with g\left(a\right)=g\left(b\right)=0. Let f_{n} be a uniformly bounded sequence of functions on \left[a,b\right]. Prove...
  6. R

    Convergence of a logaritmic series

    Homework Statement Analyze the convergence of the following series, describing the criteria used: \displaystyle\sum_{n=9}^{\infty}\frac{1}{(ln(ln(n)))^{ln(n)}} Homework Equations None The Attempt at a Solution Wolfram Alpha says it converges due to comparison test, however I can't...
  7. R

    Convergence of improper integrals theorems

    Homework Statement I'm trying to prove these two theorems a) if ## 0 \leq f(x) \leq g(x) ## for all x ## \geq 0 ## and ## \int_0^\infty g ## converges, then ## \int_0^\infty f ## converges b) if ## \int_0^\infty |f| ## converges then ## \int_0^\infty f ## converges. Obviously assuming...
  8. A

    Real Analysis Convergence Question

    1) Use mathematical induction to prove that for any k ∈ N, lim (1+k/n)^n = e^k. I already used monotone Convergence Thm to prove k=1 case. Do I just need to go through the same process to show k? If not, could you please help? 2) Suppose that ( x_n ) is a sequence of real numbers, ( y_n...
  9. S

    Convergence in measure of the product of two functions

    f_{k} \overset{m}{\rightarrow} f and g_{k} \overset{m}{\rightarrow} g over E. Then: a) f_{k} + g_{k} \overset{m}{\rightarrow} f+g over E b) If | E | < + \infty, then f_{k} g_{k} \overset{m}{\rightarrow} fg over E. Show that the hiphotesis | E | < + \infty is neccesary c) Let \{...
  10. S

    Convergence of sequences proof

    Given a sequence ## <x_n> ##, let ## <x_{n+1}> ## denote the sequence whose nth term for each ## n \in \mathbb{N} ## is ## x_{n+1} ##. Show that if ## <x_n> ## converges then ## < x_{n+1} ## converges and they have the same limit. my attempt thus far given ## \epsilon > 0 ## ##\exists N...
  11. C

    Proof of convergence

    Use only the definition of convergence to prove ## \dfrac{cos(n)}{n} \rightarrow 0 ## my proof: given ## \epsilon > 0 ## ##\exists N \in \mathbb{R} ## s.t. n>N => ## |\dfrac{cos(n)}{n}| < \epsilon ## ## |cos(n)| \leq 1 ## therefore ## \dfrac{|cos(n)|}{|n|} \leq \dfrac{1}{|n|} < \epsilon ## so...
  12. L

    Does the series converge? Exploring the convergence of ln(1+e^-n)/n

    Homework Statement So I need to determine if the series \Sigmaln(1+e^{-n})/n converges.Homework Equations The Attempt at a Solution I know it does, but cannot prove it. Wolfram says that the ratio test indicates that the series converges, but when I try to solve the limit I get that it equals...
  13. caffeinemachine

    MHB Point of convergence of a series

    Hello MHB. I have been preparing for my subject GRE and I need help on the following problem. Find $\displaystyle\sum_{k=1}^\infty \frac{k^2}{k!}$. Using the ratio test we know that the series converges but how to we find what it converges to?
  14. F

    Sequences, Series, Convergence and Divergence

    Homework Statement Q1 Are the following sequences divergent or convergent as n tends to infinity. a: \frac{5n+2}{n-1} b: tan^{-1}(n) c:\frac{2^n}{n!} Q2 Evaluate:... a: \sum_{n=1}^{\infty} 3^{\frac{n}{2}} b: \sum_{n=1}^{99} (-1)^n Q3 Find whether the following converge or diverge...
  15. A

    Real Analysis Convergence Question

    show that if a and b are distinct real numbers, then there exists a number ε > 0 such that the ε -neighorboods Vε (a) and Vε (b) are disjoint. How to solve this question? Thank you
  16. S

    The Convergence Of SOR iteration method

    Homework Statement show SOR iteration method converges for the system. $$6x+4y+2z=11$$ $$4x+7y+4z=3$$ $$2x+4y+5=-3$$ Homework Equations if the coeff. matrix is positive definite matrix and 0≤ω≤2. Then SOR converge for any initial guess. Or if $$ρ(T_{ω})$$≥|ω-1|, then SOR converge...
  17. A

    Convergence integral problem

    Homework Statement I have the attached file as an exercise for class. Problem is that I don't really understand why my book spends so much into solving it, when for me it seems pretty easy. Homework Equations Lebesgue dominated convergence theorem The Attempt at a Solution I...
  18. P

    Convergence of sequences

    Hi, Let a(n) be a real sequence such that a(n+1)-a(n) tends to zero as n approaches ∞. must a(n) converge? Also an explanation would be great thank you. have been wondering about this
  19. F

    Check for the convergence or divergence of the following series

    Homework Statement Here are some series I'm completely stuck on. 1.sqrt(n)*(1-cos(1/n)) 2. a series in which if n is odd, then an is 1/(n+\sqrt[]{n}) while if n is even, then an is -1/n Homework Equations The Attempt at a Solution For 1., I tried integral test which seemed...
  20. D

    Show convergence of this series

    Homework Statement All I want to show is that the following infinite series converges, \Sigma_{n=1}^{\inf} = \bigg(1 - n\ln\big(\frac{2n+1}{2n-1}\big)\bigg) Homework Equations Various series tests... The Attempt at a Solution I tried doing a ratio test, after applying...
  21. I

    Is the Series Ʃ n^4 / e^(n^2) Convergent?

    Homework Statement determine whether the Ʃ n4 / en2 is convergent or divergent? Homework Equations The Attempt at a Solution Using Root test: lim of n4/n / en as n approaches infinity But lim of n4/n as n approaches infinity = ∞0 So: Let N = lim of n4/n as n approaches...
  22. T

    Order of Convergence & Numerical Analysis

    Homework Statement In my book, for a class on numerical analysis, we are given the definition: "Suppose {β_{n}}from n=1 → ∞ is a sequence known to converge to zero, and \alpha_{n} converges to a number \alpha. If a positive constant K exists with |\alpha_{n} - \alpha|≤K|β_{n}|, for large...
  23. C

    MHB Sum series- convergence and divergence

    converge or diverge? \sum_{n=1}^{^{\infty }}a_{n} a_{1}= \frac{1}{3}, a_{n+1}= \sqrt[n]{a_{n}} Im having problems to solve this exercise, i would like to see your solutions
  24. O

    Blanking on word for kind of convergence of a sum

    I have a sum \sum_{n=-\infty}^{\infty} f(n) which I do not want to consider the convergence of in the normal sense, but I want to talk about the limit \lim_{N\to \infty} \sum_{n=-N}^{N} f(n). I know that when this limit exists the sum is _____ convergent, or is a _____ sum, where _____ is...
  25. I

    Convergence Test for Alternating Series and When to Use It

    Homework Statement Ʃ cos(k*pi)/k from 1 to infinity. This is a test for convergence. and when is the proper time to use the alternating series test like using it on (-1)k(4k/8k) would result to divergence since lim of (4k/8k) is infinity and not 0 but the function is really convergent...
  26. stripes

    Equivalent definitions of convergence

    Homework Statement \mathbf{D1:}\forall\varepsilon>0,\exists K\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n\geq K\Longrightarrow|x_{n}-x|<\varepsilon \mathbf{D2:}\forall\rho>0,\exists M\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n>M\Longrightarrow|x_{n}-x|\leq\rho Show these two...
  27. Seydlitz

    Another Test for Convergence Question

    Homework Statement Test the following series for convergence or divergence. $$\sum_{n=1}^{\infty} \frac{1}{3^{\ln n}}$$ The Attempt at a Solution I've tried to compare this to geometric series ##3^n## but obviously the target term is larger overall than its geometric counterpart...
  28. sr8

    Wave Convergence generator, pretty patterns

    Hi everyone, here is a web browser program of a complicated wave-pattern generator: https://dl.dropboxusercontent.com/u/114667999/Public.html i wished to have a formula that explains cymatics patterns, and patterns found in wave tanks, because they are fascinating. I wrote an program that...
  29. S

    Finding Convergence, Limits and values

    Don't really know how to get round this, the -1^n confuses me. Homework Statement Determine whether the following sequence {an} converges as n→∞? if it does, find limn→∞an Homework Equations an=(3n+(-1)n )/ (n3+2) Homework Statement
  30. W

    Convergence time of a recursive function

    I have a recursive function that will eventually converge to either a fixed value or a limit cycle. Depending on the inputs, it will converge to different values (or cycles) at different rates. How could I go about measuring the rate of convergence for different inputs, regardless of what type...
  31. F

    Convergence of Natural Log function with the limit comparison test

    Homework Statement Determine whether Ʃ(n from 1 to infinity) ln(n)/n^3 converges or diverges using the limit comparison test. Homework Equations I must use the limit comparison test to solve this problem-not allowed to use other tests. The Attempt at a Solution I know that the...
  32. L

    Prove convergence in probability for n * Poisson variable to zero

    The problem: Let \mu_{n} = \frac{1}{n} for n \in \mathbb{N}. Let X_{n} \; \mathtt{\sim} \; \textrm{ Poisson}\left( \lambda_{n} \right). Let Y_{n} = n X_{n}. Show that Y_{n} \xrightarrow{P} 0 . Work I've done: I've shown that X_{n} \xrightarrow{P} 0 by showing that \mathbb{P} \left(...
  33. M

    Explain Convergence Theorem & Contradicting Statements

    Can someone explain to me what they are saying in the paint document? Because to me it seems like the statements are contradicting. The first paragraph starts off with..." Let the fixed term be denoted..." My concern is when the paragraph states.. "If the ratio is equal to unity, each of...
  34. J

    Interval of Convergence of a Series

    Homework Statement Find the interval of convergence for the following power series. Specify both absolute and conditional convergence where appropriate.Homework Equations 1 + x + 2x^2 + 6x^3 + ... + n! x^n + ...The Attempt at a Solution Using the ratio test to determine convergence of the...
  35. F

    Does the series Ʃ sinx / x converge or diverge?

    Homework Statement Determine whether the series Ʃ(1 to infinity) sinx / x converges or diverges. Homework Equations This question appears in the integral test section, but as far as i know the integral test can only be used for decreasing functions, right? The Attempt at a...
  36. Z

    Determine the convergence or divergence of the sequence

    Homework Statement Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. an = (1*3*5*...*(2n-1))/(2n)n Homework Equations lim n->infinity an = L The Attempt at a Solution The answer in the book shows: 1/2n *...
  37. E

    Find radius of convergence and interval of convergence for the series

    x^n/(2n-1) is the series. It starts at 1 and goes to infinity. I did the ratio test on it and got abs.(x) So the radius of convergence=1, and then I plugged -1 and 1 into the original series and got that they both converged. But the answer is [-1,1). Why aren't they both hard brackets?
  38. Fernando Revilla

    MHB Binomial series (radius of convergence)

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  39. J

    Interval of Convergence for Ration Test

    Homework Statement Say that you were using ration test for ## \sum_{n=1}^\infty\frac{(-1)^{n+1} (x-4)^n}{n9^n!}\ ## Homework Equations The Attempt at a Solution You take the limit of the above you will get ##\frac {1}{9} |x-4|## Book says radius of convergence is 9...
  40. F

    Convergence rate (with unknown exact solution)

    Homework Statement I want to compute the following integral I=\int_0^1 f(x) dx for a function f(x) such that the integral I cannot be evaluated analytically. f(x) is a known function. Therefore we want to obtain I numerically. To do this we want to use the Trapezium method with...
  41. marcus

    Holonomy spinfoams: convergence of the partition function

    This looks like a significant step forward. The paper is clearly written and gives a brief historical account of progress in spin foams over the past half-dozen years or so: an understandable review that places its results in context. The authors, Hellmann and Kaminski, discover a problem with...
  42. 1

    Series-Interval of Convergence

    Homework Statement \Sigma from n=0 to infinity (x-4)^(2n)/((n+1)(11^n)) Find the Radius of Convergence Homework Equations limit of (Cn/(Cn+1) Which I found it to be 11 (isn't this supposed to be the Radius of Convergence?) The Attempt at a Solution
  43. trash

    Convergence of random variables.

    Homework Statement Given a sequence of independent random variables {X_n}, each one with distribution Exp(1). Show that Y_n = \displaystyle\frac{X_n}{\log(n)} with n \geq 2 converges to 0 in probability but it doesn't coverges almost surely to 0. Homework Equations Density for each X_n...
  44. J

    Series Convergence Verification

    Homework Statement Verify that the infinite series converges Homework Equations \Big(\sum_{n=1}^\infty\frac{1}{n(n+1)}\Big) The Attempt at a Solution So I did just that and I got 1/n -1/(n+1) So I thought to take the limit of this as n goes to infinity I got 0 as the limit. My...
  45. T

    Sequences - Definition of convergence

    Alright, I need some help with this. an = \frac{1 - 5n^{4}}{n^{4} + 8n^{3}} To find the limit of convergence, use l'Hopital's Rule. The result will come out to L = -5 From my book, "The sequence {an} converges to the number L if for every positive number ε there corresponds an...
  46. 1

    Proof of convergence (intro topology)

    Homework Statement Show that if x = (x1, x2,...) and y = (y1, y2,...) are members of l^2, then \sum^{\infty}_{i=1} |x_{i}y_{i}| Converges Homework Equations My book defines l^2 to be: { x=(x_{1}, x_{2}, ... ) \in ℝ^{\omega} : \sum^{\infty}_{i=1} (x_{i})^{2} converges }...
  47. Z

    Calculus: Absolute/Conditional Convergence or Divergence Question

    Homework Statement This is the problem: http://i.imgur.com/AtISqng.jpg its the first series, not the second one with the cos Homework Equations The Attempt at a Solution so anyways i did this question but i just had one doubt. in every other question like this that I've done, the series...
  48. C

    Alternating Series Convergence Test

    According to my calculus book two parts to testing an alternating series for convergence. Let s = Ʃ(-1)n bn. The first is that bn + 1 < bn. The second is that the limn\rightarrow∞ bn = 0. However, isn't the first condition unnecessary since bn must be decreasing if the limit is zero. I...
  49. B

    Convergence of this sequence .

    Homework Statement find the limit n\rightarrow∞ of 10n/ n! Homework Equations L hospital rule The Attempt at a Solution took log and separated the num and denom as: n ln10-ln(n!) n ln10-n ln(n)+n 1/n ( ln10 - ln(n)+1) now i...
  50. D

    Ratio test for math convergence

    Homework Statement show ## \sum \frac{x^{2}}{(1+x^{2})^{n}} ## converges uniformly on R Homework Equations The Attempt at a Solution I know by ratio test it is absolutely convergent for all x in R. I am guessing you use m-test. However I do not really understand how...
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