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. P

    Convergence Doubt: Is Most Repeated Value the Answer?

    Hello people, In case if I am typing the question in the wrong forum please redirect me. OK, here it goes... I have this stupid question: Suppose we have a sequence 1, 1, 1, 1, 1, 1... It converges to '1'. Consider 1, 0, 1, 0, 1, 0... it diverges right? What about a sequence 1, 1, 0, 1, 1...
  2. N

    Question: Super Convergence and Convergence: Is There a Relationship?

    Homework Statement Suppose (## a_n ##) is a sequence and let l\in\mathbb R. Let us say that (## a_n ##) is "super convergent" to ##l## if there exists N\in\mathbb N such that for every ε>0 we have ##n \geq N## ⇒ |(## a_n - l##|<ε . Show that if (## a_n ##) super converges to l then (## a_n...
  3. S

    Convergence and divergence of a series

    B]1. Homework Statement [/B] Find whether the series is convergent or divergent Homework Equations The Attempt at a Solution By ratio test I have, I would apply L'Hôpital's rule to find the value of limit but before that how do i simplify the expression? It has fractional part both in the...
  4. C

    Convergence of infinite sequences

    Homework Statement Let V consist of all infinite sequences {xn} of real numbers for which the series summation xn2 converges. If x = {xn} and y = {yn} are two elements of V, define (x,y) = summation (n=1 to infinity) xnyn. Prove that this series converges absolutely. Homework Equations The...
  5. E

    Series Convergence: Explaining P>1 & P>0

    Homework Statement Hi, everyone. I'd appreciate it if someone could explain something for me regarding the convergence of series. Thanks in advance![/B] Homework Equations In my calculus book, I'm given the following: (1) - For p > 1, the sum from n=1 to infinity of n^-p converges. (2) -...
  6. A

    MHB Improper Integral Question (convergence & evaluation)

    Hello, Two questions will be posed here. (1) Question about Convergence; quick way. Hello, I am trying to learn this concept on my own. My major question here is that, Is there a quick way, to tell if an integral converges or diverges? Suppose $\int_{0}^{\infty} \frac{x^3}{(x^2 +...
  7. B

    Pointwise convergence of Riemann integrable functions

    Hello Normally in order to change the order of limit and integration in rimann integration, you need uniform convergence. But let's say that you are not able to prove uniform convergence, but only pointwise convergence. And let's say you are able to prove that the functions are also...
  8. Julio1

    MHB What is the Radius of Convergence for the Power Series?

    Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$
  9. R

    MATLAB Solving Steady State Heat Conduction Eqn w/ MATLAB

    HI guys,this is my first programming experience , i have developed an MATLAB code for steady state heat conduction equation , on governing equation dt2 /dx2 + dt2/dy2 = -Q(x,y) i have solved this equation with finite difference method, As far as i know if we increase the mesh size it leads...
  10. C

    Definition of Convergence: Can n -> -infinity

    Homework Statement I've been given a question that makes use of 5^(n)*sin(pi*n!) The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity? So basically, I...
  11. I

    Convergence of iterative method and spectral radius

    Homework Statement Show that if given \mathbf{x}_0, and a matrix R with spectral radius \rho(R)\geq 1, there exist iterations of the form, \mathbf{x}_{n+1}=R\mathbf{x}_0+\mathbf{c} which do not converge. The Attempt at a Solution Let \mathbf{x}_0 be given, and let...
  12. M

    MHB The proof of local convergence of the following function....

    We work on project about mobile cellular networks. In a part of the project, we faced the problem about the proof of convergence of ...(Complete description is Available in attachment.Please download it) Please help me.It is very important!
  13. K

    Power series absolute convergence/ Taylor polynomial

    1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...
  14. I

    MHB Determine Convergence/Divergence: Series Answers

    my final is tomorrow and my instructor gave a list of questions that will be similar to the ones on the final exam and i want to see how they should be done properly. I've been working on other problems but i can't get past these ones. thanks determine convergence/divergence...
  15. I

    MHB Taylor Series Expansion and Radius of Convergence for $f(x)=x^4-3x^2+1$

    find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?
  16. I

    MHB What are the intervals and radius of convergence for two series with typos?

    find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$
  17. Chacabucogod

    Fourier Series Convergence Criterion

    I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States: " If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...
  18. S

    Convergence of perturbative solutions to a non-linear diff eq

    Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it. We have the equation \begin{equation} \partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right) \end{equation} We will look for solutions satisfying...
  19. Greg Bernhardt

    How Do Series Converge in Normed Spaces?

    Definition/Summary In what follows, we will work in a normed space (X,\|~\|). A series is, by definition, two sequences (u_n)_n and (s_n)_n such that s_n=\sum_{k=0}^n{u_k} for every n. We call the elements u_n the terms of the series. The elements s_n are called the partial sums. We will...
  20. M

    Finding the Radius of Convergence for a Series with a Real Coefficient

    Hello. How do I find the radius of convergence for this problem? ##\alpha## is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$ I understand that we can use the ratio test to find R. And by using ratio test, I got R=1. But in the...
  21. A

    MHB What is the Radius of Convergence for a Series with a Real Non-Zero Alpha?

    Hello. How do I find the radius of convergence for this problem? $\alpha$ is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$
  22. M

    Macluarin series and radius of convergence

    Hello. I am stuck on this question. I'd appreciate if anyone could help me on how to do this. The question: Expand the following into maclaurin series and find its radius of convergence. $$\frac{2-z}{(1-z)^2}$$ I know that we can use geometric series as geometric series is generally...
  23. F

    MHB Why Does Choosing z=1/n Demonstrate Non-Uniform Convergence?

    on page 4, example 9 in this link, http://www.personal.psu.edu/auw4/M401-notes1.pdf, they show a sequence of functions is not uniformly convergent. To show this, you need to show that for some epsilon, there is no 'universal' N. But they didn't pick a particular value of $z$, they chose...
  24. M

    Finding the Radius of Convergence for $zsin(z^2)$ in Maclaurin Series

    Hello. I need explanation on why the answer for this problem is R = ∞. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$...
  25. A

    MHB What is the Radius of Convergence for zsin(z^2)?

    Hello. I need explanation on why the answer for this problem is $R=\infty$. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$ Divide...
  26. A

    MHB Series Convergence Test Questions

    Just a few quick questions this time: I'm doubting the first one mostly, because when I used the integral test to evaluate it: I ended up getting (-1/x)(lnx +1) from 2 to infinity, which gave me an odd expression: (-1/infinity)(infinity +1 -ln2 -1). I'm assuming this means it is convergent...
  27. A

    MHB Sequence Divergence and Convergence Questions

    Hey guys, I have a couple more questions. For the first one, taking the limit to infinity obviously equals 0 so it should be convergent, right? Also, for the second one, the limit as n approaches infinity for gives me indeterminate form, so I took the derivative which just gave me ln(n)...
  28. A

    MHB Series Convergence and Divergence III

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm highly doubtful of my answer for c. I used the roots test instead of the ratio test, which gives 1/n, which I took the limit of to get an interval of [-∞ , ∞] As for a and b, I got [-5,5] and (-∞, ∞)...
  29. A

    MHB Series Convergence and Divergence II

    Hey guys, I have a few more questions for the problem set I'm working on at the moment: I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute...
  30. A

    MHB Series Convergence and Divergence I

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm mostly unsure of my response for b. For a, I just split the series into two parts and added 6+3 to get 9, and thus the series is convergent. For c, I got 3/5 after taking the limit, which is...
  31. A

    MHB Integral Convergence and Divergence II

    This thread is only for question 5. As for number 5 part a, after tediously expanding the partial fraction expression, I ended up getting c=1, d=0, b=1, and c=1, ultimately resulting in: ln(x) - (1/x^2) + c. I really don't think this looks right. As for 5b, I obtained b=-1, c=-1, a=2, and...
  32. A

    MHB Calculus II Integral Convergence and Divergence Questions

    For a,b, and c respectively, I got divergent (to -infinity), convergent (to π/6), and divergent (to infinity, since the first part's sum is 1/3, but lim negative infinity gives infinity, thus the summation of the two integrals gives a divergent integral). I'm sure these are right, but I'd...
  33. A

    MHB Integral Convergence and Divergence I

    Hello, I'm doubting a couple of my answers for these questions. Some of them seem relatively simple, but there are slight nuances that I'm not sure of. This thread is only for question 4. For 4a, I just used the (a^2) - (x^2) => x=asin(Ø) rule and substituted 3sin(Ø) for x. I ended up...
  34. karush

    MHB Series Convergence: Test for x Values

    (x-1)-\frac{(x-1)^2}{2!}+\frac{(x-1)^3}{3!}-\frac{(x-1)^4}{4!}+ ∙ ∙ ∙ well this looks like an alternating-series, the question is: at what value(s) of x does this converge. one observation is that if x=0 then all terms are 0 so there is no convergence, also I presume you can rewrite this as...
  35. G

    Series convergence / divergence

    Homework Statement Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally? \sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n}) Homework Equations Alternating series test The Attempt at a Solution I started out by taking the limit of ##a_n...
  36. A

    Nested sequence of closed sets and convergence in a topological space.

    Homework Statement Let ##X## be a topological space. Let ##A_1 \supseteq A_2 \supseteq A_3...## be a sequence of closed subsets of ##X##. Suppose that ##a_i \in Ai## for all ##i## and that ##a_i \rightarrow b##. Prove that ##b \in \cap A_i##. Homework Equations The Attempt at a Solution...
  37. Y

    Series convergence for certain values of p

    Homework Statement For which integer values of p does the following series converge: \sum_{n=|p|}^{∞}{2^{pn} (n+p)! \over(n+p)^n} Homework Equations The Attempt at a Solution I'm trying to apply the generalised ratio test but get down to this stage where I'm not sure what...
  38. G

    Convergence / Divergence of a series

    Homework Statement Does the following series converge or diverge? ##∑\frac{n^5}{n^n}## (as n begins from 1 and approaches infinity) Homework Equations Ratio test? The Attempt at a Solution For your reference, thus far I have learned about the geometric series, the limit test...
  39. ChrisVer

    Convergence of q-series for rational x values

    Homework Statement For 0<q<∞, and x rational, for what x values does the series converge? \sum_{n=0}^{∞} q^{1/n} x^nThe Attempt at a Solution I don't know which method works best for this
  40. W

    A logarithmic convergence tests - Analysis

    Homework Statement Given a non-negative sequence \{a_{n}\}_{n=1}^{\infty}. Proove that the serie \Sigma_{n=1}^{\infty}a_{n} converge if and only if \Sigma_{n=1}^{\infty}\ln(1+a_{n}) converges. Homework Equations The Attempt at a Solution My first attempt is the direct...
  41. M

    Convergence of Series: Check a) & b)

    Homework Statement Check if the series below converge. a) $$\sum_{n = 1}^\infty \frac{n}{2n^2 - 1}$$ b) $$\sum_{n = 2}^\infty (-1)^n \frac{2n}{n^2 - 1}$$ Homework Equations The Attempt at a Solution For a). The series converge if the sum comes up to a finite value. If...
  42. N

    Convergence of MGF (looking for proof)

    I'm looking for the proof of the following theorem/statement. $$ \begin{align} \lim_{n \to \infty} M_{X_n}(t) = M_X(t) \end{align} $$ for every fixed t \in \mathbb{R}. My book only states the theorem without proving it and I haven't found a proof online. Any help is appreciated! :)
  43. MarkFL

    MHB The Royal Primate's question at Yahoo Answers regarding convergence of a series

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  44. M

    MHB Prove $\lim_{n \to \infty} \sqrt[n]{a_1a_2...a_n} = L$ for Convergent ${a_n}$

    Assume the sequence of positive numbers ${a_n}$ converges to L. Prove that $\lim_{n \to \infty} \sqrt[n]{a_1a_2...a_n} = L$ (The nth root of the product of the first n terms) Since ${a_n}$ converges we know that for every $\epsilon> 0$ there is an $N$ such that for all $n > N$ $ |a_n -...
  45. Fredrik

    Does L^1 (seminorm) convergence imply a.e. convergence?

    Define ##\rho(f)=\int |f|\mathrm d\mu## for all integrable ##f:X\to\mathbb C##. This ##\rho## is a seminorm, not a norm. Does ##\rho(f_n-f)\to 0## imply ##f_n\to f## a.e.? I kind of think that it should, because in the case of real-valued functions, ##\int|f_n-f|\mathrm d\mu## is the area...
  46. B

    Proving the Monotone Convergence Theorem for Non-Negative Measurable Functions

    Homework Statement Let f be a non-negative measurable function. Prove that \lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.The Attempt at a Solution I feel like I'm supposed to use the monotone convergence theorem. I don't know if I'm on the right track but I created a...
  47. Z

    Ansys frictional contact - convergence problem

    Hi, I am writing to you to ask for help. I have a problem with the contact "Frictional". I'll show you screenshots of my settings and program the console errors. Everything will be shown in the screenshots. The problem is that between the beams and the top element and the bottom element is...
  48. H

    Convergence of Series: Exploring Divergence in Negative Series

    Homework Statement I know that the harmonic series is divergent.But why \frac{-1}{n} is also divergent? I've search for some test to test that, but I could not find a method for negative series. So how can I prove the series is divergent?:confused: Homework Equations The Attempt...
  49. L

    Behaviour of series (radius of convergence)

    Homework Statement Series: \sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n} what is the behaviour of the series at radius of convergence \rho_o=-z ? Homework Equations The Attempt at a Solution So I can specify that the series is monatonic if z is non negative as...
  50. L

    Radius of convergence log(a + x)

    Homework Statement determine the radius of convergence of the series expansion of log(a + x) around x = 0 Homework Equations The Attempt at a Solution So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the...
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