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. E

    Use the Limit Comparison Test to determine the series' convergence?

    Homework Statement Use the Limit Comparison Test to determine if the series converges or diverges: Ʃ (4/(7+4n(ln^2(n))) from n=1 to ∞. (The denominator, for clarity, in words is: seven plus 4n times the natural log squared of n.) Homework Equations Limit Comparison Test: Let Σa(n) be the...
  2. F

    MHB Convergence of bounded linear operators

    Let (T_{n}) be a sequence in {B(l_2} given by T_{n}(x)=(2^{-1}x_{1},...,2^{-n}x_{n},0,0,...). Show that T_{n}->T given by T(x)==(2^{-1}x_{1},2^{-2}x_{2},0,0,...). I get a sequence of geometric series as my answer for the norm, but not sure whether that's correct.
  3. X

    Radius and interval of convergence help

    I have a problem: the radius of convergence for the power series \sum(-1)^n \frac{(x-3)^n}{(n+1)} is R=1. Determine the interval of convergence. What does this mean? can anyone help me solve this please
  4. A

    Convergence of Positive Sequences: Limits and Sums from 1 to Infinity

    Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity. My approach is that lim a_n/b_n = 0 means that there exists N in natural number (N) for...
  5. S

    Proof of subsequence convergence

    prove if ## a_{2k} \rightarrow l ## and ## a_{2k-1} \rightarrow l ## then ## a_n \rightarrow l ## where ## a_{2k} ## and ## a_{2k-1} ## are subsequences of ## a_n ## my attempt: since: ## a_{2k} \rightarrow l ## then ## \forall \epsilon > 0 ## ##\exists N_1 \in \mathbb{R}## s.t. ##2k > N_1...
  6. A

    Determine the values of x for series convergence

    Homework Statement Determine the values of x for which the following series converges. Remember to test the end points of the interval of convergence. ^{∞}_{n=0}\sum\frac{(1-)^{n+1}(x+4)^{n}}{n} Homework Equations I worked it down to |x+4|<1 ∴-5<x<-3 The Attempt at a Solution...
  7. L

    Uniform Convergence Homework: Is ##f_n(x) = \frac{x}{1+nx^2}##?

    Homework Statement Is the sequence of function ##f_1, f_2,f_3,\ldots## on ##[0,1]## uniformly convergent if ##f_n(x) = \frac{x}{1+nx^2}##? 2. The attempt at a solution I got the following but I think I did it wrong. For ##f_n(x) = \frac{x}{1+nx^2}##, I got if ##f_n \to0## then we must...
  8. Sudharaka

    MHB Neighbourhood of Convergence of Sequence

    Hi everyone, :) Can somebody give me a hint to solve this problem. :) Problem: Let \(f\) be a function defined on \([a,\,b]\) with continuous second order derivative. Let \(x_0\in (a,\,b)\) satisfy \(f(x_0)=0\) but \(f'(x_0)\neq 0\). Prove that, there is a neighbourhood of \(x_0\), say...
  9. L

    Uniform Convergence of ##\{f_n\}## on ##[0,a]##

    Homework Statement Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.2. The attempt at a solution Is this correct? Should I add more detail? Thanks for your help! Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to...
  10. J

    Convergence of different macromolecular structures with same function

    My professor had given us a lecture on the RNA World Hypothesis. He provided evidence for this hypothesis by citing the in-vitro selection experiments carried out on RNA. The outcome of one (or more) of these experiments resulted in the "creation" of an RNA polymerase ribozyme which totally...
  11. Z

    Convergence of a Recursive Sequence

    Homework Statement The following sequence comes from the recursion formula for Newton's Method. x0= 1 , xn+1=xn-(tanxn-1)/sec2xn Show if the sequence converges or diverge. Homework Equations The Attempt at a Solution I don't really know where to start on this problem, I have tried to use some...
  12. 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
  13. A

    Does the Series \(\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}\) Converge or Diverge?

    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
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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 \{...
  21. 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...
  22. C

    Does Cos(n)/n Converge to Zero as n Approaches Infinity?

    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...
  23. 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...
  24. 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?
  25. 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...
  26. 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
  27. 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...
  28. A

    Why Is My Approach to the Lebesgue Dominated Convergence Theorem Incorrect?

    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...
  29. P

    Does a Sequence Converge if Its Differences Tend to Zero?

    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
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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
  41. 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...
  42. 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...
  43. 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(...
  44. 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...
  45. 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...
  46. 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...
  47. 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 *...
  48. 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?
  49. 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.
  50. 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...
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