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. M

    Find the interval of convergence for the given power series.

    Homework Statement Find the interval of convergence for the given power series. Sum from n=1 to infinty of (x-11)^n / (n(-9)^n) Homework Equations The Attempt at a Solution I used the ratio test and I'm getting 2<x<20, but that doesn't seem to be right. I get abs(1/9*(x-11)) <...
  2. S

    Does the series converge or diverge? (r=1..inf)

    \sum (1-\frac{1}{r})^{r^2} Does this converge or diverge.(r=1..inf) I have tried the following but do not think it is adequate(or correct for that matter) (1-\frac{1}{r})^r (1-\frac{1}{r})^r = (1-\frac{1}{r})^{r^2} and lim (1-\frac{1}{r})^r -> \frac{1}{e} thats given from a...
  3. S

    Convergence of Series (Harder)

    Prove that: (1-\frac{1}{n})^n \rightarrow \frac{1}{e} as n \to \infty you may use the fact that (1+\frac{1}{n})^n \rightarrow e I have no idea where to even begin, can someone point me in the right direction ?
  4. DryRun

    Convergence of Series: Is There Only One Way to Solve This?

    Homework Statement \sum_{n=0}^{\infty}\frac{1}{n^2+3n+2} The attempt at a solution I'm wondering if there is only one way of solving this? Here is what I've done: First, converting into partial fractions. Is there a way to do it without converting to partial fractions...
  5. M

    Convergence of a Series: How to Determine its Value?

    Homework Statement I'm not sure how to do the notation on here but. Does this series converge or diverge. If it converges, then to what value. The series: Sum from 1 to infinity of [(-1)^n * n / (n^2-4n-4)] Homework Equations It tells me to use the ratio test The Attempt at a...
  6. M

    Convergence of {n/(n^2+1)}: Is it Possible?

    Homework Statement Is the sequence {n/(n^2+1)} convergent, and if so, what is it's limit?Homework Equations The Attempt at a Solution I believe it does converge because the higher power is in the denominator, so thus, it's limit is 0. Any help or hints on if I'm headed in the right direction...
  7. T

    Determining General Values of Convergence for a seqence

    Determine the values of "r" for which rn converges. Is there a specific procedure I should try to apply to figure this out? The only things I could intuitively come up with that will converge in this scenario are when -1 ≤ r ≤ 1...is there anything else to this?
  8. G

    Finding the Annulus of Convergence for a Laurent Series

    I am trying to understand the idea of annulus of convergence. This is the example I have been looking at but it has me completely stumped. [∞]\sum[/n=1] (z^n!)(1-sin(1/2n))^(n+1)! + [∞]\sum[/n=1] (2n)!/[((n!)^2)(z^3n)] All of the examples I have worked on in the past have been...
  9. T

    Why can a term be factored out of the numerator in a converging sequence?

    State whether the sequence converges and if so, find the limit (n+1)1/2/2(n)1/2 ok so I got that it converges to 1/2, my question more so lies in the fact that why are we able to factor out a (n)1/2 from the term in the numerator? Isn't it only the denominator that we are concerned about...
  10. D

    Weierstrass Product convergence

    Show that the infinite product f(z) = \prod\limits_{n = 0}^{\infty}(1 + z^{2^n}) converges on the open disc D(0,1) to the function 1/(1 - z). Is this convergence uniform on compact subsets of the disc? This should actually be done by the comparison test. For |z| < 1, we have that $$...
  11. S

    Convergence of sequence of measurable sets

    Given a totally finite measure μ defined on a \sigma-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [A_\alpha] where...
  12. D

    MHB Show convergence of weierstrass product

    $f(z) = \prod\limits_{n=1}^{\infty}\left(1+z^{2^n}\right)$ converges on the open disc $D(0,1)$ to the function $\dfrac{1}{1-z}$. To show convergence, I look at $$ \sum_{n=1}^{\infty}\left|z^{2^n}\right| $$ correct?The sum, $\sum\limits_{n = 0}^{\infty}|z|^{2^{n}}$, converges for $|z| < 1$ i.e...
  13. T

    Monotone Convergence Theorem Homework: Integrals & Increasing Sequences

    Homework Statement Homework Equations Monotone Convergence Theorem: http://img696.imageshack.us/img696/5469/mct.png The Attempt at a Solution I know this almost follows from the theorem. But I first need to write \displaystyle \int_{I_n} f = \int_S f_n for some f_n in such a...
  14. H

    Convergence of improper integrals with parameters

    I'm having a lot of trouble with the subject. Here's one example I'd like explained. F(t_1, t_2) = \int \limits_0^1 x^{t_1}\ln^{t_2}\frac{1}{x} dx The book asks to find for what \vec{t} F converges. The answer is \vec{t}\in(-1; \infty)^2, but I don't see how to get that. In general, what...
  15. E

    Understanding Uniform Convergence: The Role of N and A

    Homework Statement I would just like to be pointed in the right direction. I have this theorem: Let E be a measurable set of finite measure, and <fn> a sequence of measurable functions that converge to a real-valued function f a.e. on E. Then given ε>0 and \delta>0, there is a set...
  16. M

    Help me find the radius of convergence?

    Homework Statement Ʃn!(x-1)n I need to find the radius of convergence for this summation from n=0 to n=∞ The Attempt at a Solution I started off with the ratio test: (n!(n+1)(x-1)(x-1)n)/(n!(x-1)n) = (n+1)(x-1) (x-1)lim(n+1)...Now at this point it looks to me like the series does...
  17. N

    Determining the convergence or divergence with the given nth term

    [answered] I want to know why this particular approach is wrong so I can learn from my mistakes. Homework Statement a_n = \frac{ln(n^3)}{2n}The Attempt at a Solution For the sake of being time efficient, I will skip writing things like the limit as n approaches infinity etc. a_n =...
  18. T

    The Convergence of Complex Integrals

    I know that for any a>0 and k,t\in\mathbb{R}, the integral \int_0^a t^k\; dt converges if and only if k>-1. Is it true that if k is complex then \displaystyle \int_0^a |t^k| \; dt converges if and only if \text{Re}(k)>-1 since if t is real, |t^k| does not depend on the imaginary part of k?
  19. T

    Absolite convergence of the beta function

    Determine the values of the complex parameters p and q for which the beta function \int_0^1 t^{p-1} (1-t)^{q-1}dt converges absolutely. The solution says: Split the integral into 2 parts: \displaystyle \int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1...
  20. I

    True/false convergence of integral from 1 to infinity

    Suppose h(x) is a continuous function for x > 0. If \int^∞_1{h(x)dx} converges then for constant 0 < a < 1, \int^∞_1{h(\frac{x}{a})dx} also converges. The answer is true. Anyone care to explain why? I would have chosen false, because I was thinking that h(x/a) is larger than h(x) so we...
  21. M

    Calculus 2, Series Convergence Questions?

    I have some problems here with Series and Convergence... Here are the problems and my guesses at it. http://img822.imageshack.us/img822/9523/23341530.png It won't tell me which one is wrong, but it just says one/all is wrong. Any help is appreciated. Attempts at solving, I tried...
  22. L

    MATLAB A convergence problem when i change a parameter in an algorithm wrote in MatLab

    Hello! I use the Crank-Nicolson numerical method to calculate temperatures on a metalic plate(aluminium) 10cmx30cm. I assume that the left vertical side is always in 0 degrees Celsious and the right vertical side in 100 degrees.Both horizontal sides are insulated. So we have a heat...
  23. A

    Finding Region of convergence for complex series

    Homework Statement I am struggling to answer this question please help Find the region of convergence for the following complex series and draw the region Ʃ(i+z)^(2n-1)/2^(2n+1)The Attempt at a Solution Here is my hand written working, sorry i could figure out how to use the symbols...
  24. S

    Improper integral convergence and implications of infinite limits

    Homework Statement Let f be a continuous function on [1,∞) such that \lim_{x\rightarrow ∞}f(x)=α. Show that if the integral \int^{∞}_{1} f(x)dx converges, then α must be 0. Homework Equations Definition of an Improper Integral Let f be a continuous function on an interval [a,∞). then we...
  25. A

    Convergence in L^2 Norm: Understanding Subsequence Implications

    Suppose there exists a sequence f_n of square-integrable functions on \mathbb R such that f_n(x) \to f(x) in the L^2-norm with x \ f_n(x) \to g(x), also in the L^2-norm. We know from basic measure theory that there's a subsequence f_{n_k} with f_{n_k}(x) \to f(x) for a.e. x. But my professor...
  26. L

    Radius of Convergence of power series anx^n^2

    Homework Statement Suppose that the power series \sumanxn for n=0 to n=∞ has a radius of convergence R\in(0,∞). Find the radii of convergence of the series \sumanxn2 from n=0 to n=∞ and \sumanx2n.Homework Equations Radius of convergence theorem: R = 1/limsup|an|1/n is the radius of...
  27. M

    Sequences and convergence in the standard topology

    Hello all. I have to present a proof to our Intro to Topology class and I just wanted to make sure I did it right (before I look like a fool up there). Proposition Let c be in ℝ such that c≠0. Prove that if {an} converges to a in the standard topology, denoted by τs, then {can}...
  28. H

    Proving convergence of 1/log(n)

    Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)
  29. H

    Determining Convergence of ((sin(n))^4)/(1+n^2)

    Suppose I want to determine the convergence of ((sin(n))^4)/(1+n^2) using limit comparison test. I divide it by 1/(1+n^2). All that remains is (sin(n))^4. Now as the limit goes to infinty, the range of values (sin(n))^4 can give is 0 to 1. Now it gives many more values above zero then at zero...
  30. T

    Radius of convergence question

    Homework Statement Suppose that the following series converges when x = -4 and diverges when x = 6. ∑{n=0 -> ∞} c_n • x^n What is the interval of convergence? The Attempt at a Solution I think it is [-5,5) but my friend reckons that it is [-5,6). I don't think [-5,6) is correct because this...
  31. S

    Convergence /Divergence of series:sec(n)/n

    I am wondering if this is solvable. Determine the convergene/divergence of the sum from n=1 to infinity of sec(n)/n. All the tests appear to fail and listing out the sequence of partial sums produces no useful results.
  32. W

    Convergence radius of a perturbation series

    i see people discussing the convergence radius of a perturbation series in the literature i am really baffled generally, one can only get the first few coefficients of a perturbation series that is, the perturbation series is not known at all how can one determine the convergence...
  33. K

    Convergence of Sequence: (n^2)/(e^n)

    NEVERMIND! IT IS 0! I SOMEHOW WAS STARING AT THE WRONG ANSWER SHEET FOR A LITTLE BIT! THANK YOU! 1. Homework Statement Determinte whether the sequence converges or diverges: (n^2)/(e^n)2. Homework Equations The book says that the solution is: e/(e-1). However, the limit of the equation...
  34. J

    Example of Sequence with Order of Convergence 5 to 3

    Homework Statement Give an example of a sequence which converges to 3 with order of convergence 5. Prove your answer. Homework Equations Order of convergence: {pn} converges to p with order \alpha. limn\rightarrow\infty |pn+1 - p|/|pn-p|\alpha = \lambda The Attempt at a Solution...
  35. A

    Convergence of a recurrence equation: x(k+1) = 0.5x(k) + u(k)

    Homework Statement Hello. I am trying to prove a result that I have been making use of, but never really proved. Consider the recurrence equation x(k+1) = 0.5 x(k) + u(k), where u(k) is a bounded sequence. For this problem, assume that u(k) goes to zero. I want to prove that x(k) goes to...
  36. H

    Uniform Convergence of Fourier sine and cosine series

    Homework Statement f(x)= {1, ‐1/2<x≤1/2} {0, ‐1<x≤ ‐1/2 or 1/2<x≤1} State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞ Homework Equations The Attempt at a Solution...
  37. T

    Using ratio test to test conditional convergence?

    Homework Statement So what I was taught was that if the lim of the ratio test is the series is always absolutely convergent. If it is >1 the series is always divergent. But if it is =1 then we don't know. So would that mean that all conditionally convergent series would have a limit = 1? I...
  38. polydigm

    Convergence of complex log series on the boundary

    The radius of convergence of \sum\limits_{k=1}^\infty\displaystyle\frac{z^n}{n} is 1. It converges on all of the boundary \partial B(0,1) except at z=1. One way of looking at this is to analyse \sum\limits_{k=1}^\infty\displaystyle\frac{\cos n\theta}{n}+\frac{\sin n\theta}{n}. You can see the...
  39. M

    MHB Fourier series, pointwise convergence, series computation

    Let $f(x)=-x$ for $-l\le x\le l$ and $f(l)=l.$ a) Study the pointwise convergence of the Fourier series for $f.$ b) Compute the series $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)}.$ c) Does the Fourier series of $f$ converge uniformly on $\mathbb R$ ? ------------- First I need to...
  40. T

    Integral Test: What should I compare this series with to prove it's convergence?

    Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/HW%20Pictures/integraltest.PNG The Attempt at a Solution I know it is convergent because it is very similar to 1/n^1.5 which is convergent as well. However what would I compare this with using the Integral Test to...
  41. H

    MHB Find the Conditions on A for Convergence of f(x) Root

    Hey guys, I can't get his question dealing with orders of convergence at all so any help would be nice. Q: Find the conditions on A so that the iteration $$x_{n+1}=x_n-Af(x_n)$$ will converge to a root of f if stared near the root. I know I should look at the taylor series expansion of f about...
  42. L

    Absolute and conditional convergence of series (1 + x/n)^n^2

    Homework Statement \sum from n=1 to n=\infty (1 + \frac{x}{n})n2 Determine the values of x for which the series converges absolutely, converges conditionally and diverges.The Attempt at a Solution So i tried using the root test for the absolute value of (1 + \frac{x}{n})n2, but it was...
  43. N

    Need to find if a sequence of functions has uniform convergence

    Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...
  44. B

    Proving Convergence in C[0,1] with Integral Norm

    Homework Statement Show that a sequence ##f_n \to f \in C[0,1]## with the sup norm ##|| ||_\infty##, then ##f_n \to f \in C[0,1]## with the integral norm. The Attempt at a Solution given ##\epsilon > 0 \exists n_0 \in N## s.t ##||(fn-f) (x)|| < \epsilon \forall n > n_0## with ##...
  45. A

    Showing X Vector Concentrated Around Unit Sphere Convergence of Mean-Squared

    Homework Statement Let Y_i be standard normal random variables, and let X be an N vector of random variables, X=(X_1, ..., X_N) where X_i = 1/{sqrt{N}} * Y_i. I want to show that as N goes to infinity, the vector X becomes "close" to the unit sphere. Homework Equations The...
  46. S

    Trying to find the radius of convergence of this complicated infinite series

    Homework Statement k is a positive integer. \sum^{\infty}_{n=0} \frac{(n!)^{k+2}*x^{n}}{((k+2)n)!} Homework Equations The Attempt at a Solution I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get...
  47. S

    Trying to find the interval of convergence of this series: run into a problem

    Homework Statement The series: \sum^{n=\infty}_{n=0} \frac{(-1)^{n+2}(x^{3}+8)^{n+1}}{n+1} Homework Equations The Attempt at a Solution using the ratio test, I get the following: |x^{3}+8|<1, but I know that the radius of convergence must be in the form: |x-a|<b, where...
  48. S

    Trying to find interval of convergence for a geometric series

    Homework Statement here is the series: \sum^{\infty}_{n=0}x(-15(x^{2}))^{n} Homework Equations The Attempt at a Solution I know that -1<-15x^{2}<1 for convergence (because of geometric series properties) but I run into a problem here: -1/15<x^{2}<1/15 You can't...
  49. D

    Convergence and stability in multivariate fixed point iteration

    Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...
  50. H

    Which Method to Use for Testing Convergence in Integrals with Substitution?

    Homework Statement Use integration, the direct comparison test, or the limit comparison test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. Homework Equations ∫sinθdθ/√π-) The Attempt at a Solution I don't know which method...
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