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. C

    Finite element solving of Laplace's equation doesn't converge

    Homework Statement I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
  2. S

    What is the Interval of Convergence for the Series ##\sum\frac{n^n}{n!}z^n##?

    Homework Statement Find the interval of convergence of: ##\sum\frac{n^n}{n!}z^n## Homework EquationsThe Attempt at a Solution I obtained that the radius of convergence is ##1/e## but I am not sure what to do at the end points. For ##z=1/e## I would have ##\sum{n^n}{n!e^n}##. Mod edit: I think...
  3. S

    Convergence of a double summation using diagonals

    Homework Statement Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##. Homework Equations I've included some relevant information below: The Attempt at a Solution So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
  4. E

    MHB Prove limit with convergence tests

    I need to prove that the limit of the sequence is as shown(0): 1.limn→∞ n*q^n=0,|q|<1 2.limn→∞ 2*n/n! but I need to do this using the convergence tests. With the second sequence I tried the "ratio test", and I got the result limn→∞ 2/n+1 which means that L in the ratio test is 0 and so it...
  5. Delta2

    I Rational sequence converging to irrational

    In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a...
  6. S

    Ratio Test and Radius of Convergence for ∑ ((n-2)2)/n2, n=1: Homework Solution

    Homework Statement ∞ ∑ = ((n-2)2)/n2 n=1 Homework Equations The ratio test/interval of convergence The Attempt at a Solution **NOTE this is a bonus homework and I've only had internet tutorials regarding the ratio test/interval of convergence so bear with me) lim ((n-1)n+1)/(n+1)n+1 *...
  7. T

    MHB Power Series Convergence Assistance

    The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.
  8. R

    Proving the convergence of series

    Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
  9. L

    A Convergence order of central finite difference scheme

    For example, when we solve simple 1D Poisson equation by finite difference method, why three point central difference scheme on uniform grid (attached image) is second order method for solution convergence? I understand why approximation of first derivative is second order (and that second...
  10. T

    Refraction Convergence and Amplitude change- Ocean waves

    There are many explanations on the internet, of refraction and convergence of ocean waves entering shallow water around a headland However they all go no deeper than this statement "Where the water is shallow the wave rays converge wave energy is greater where the wave rays spread out the...
  11. Euler2718

    Proof of sequence convergence via the "ε-N" definition

    Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...
  12. A

    MHB What point does the spiral converge to?

    Starting from the origin, go one unit east, then the same distance north, then (1/2) of the previous distance west, then (1/3) of the previous distance south, then (1/4) of the previous distance east, and so on. What point does this 'spiral' converge to? I have attempted to sketch this out but...
  13. L

    A Gamma function convergence of an integral

    ##\Gamma(x)=\int^{\infty}_0 t^{x-1}e^{-t}dt## converge for ##x>0##. But it also converge for negative noninteger values. However many authors do not discuss that. Could you explain how do examine convergence for negative values of ##x##.
  14. G

    Electric field integral: Convergence where ρ is nonzero

    Hi. I know how to use Gauss' Law to find the electric field in- and outside a homogeneously charged sphere. But say I wanted to compute this directly via integration, how would I evaluate the integral...
  15. Julio1

    MHB Convergence in topological space

    Let $(X,\tau)$ an topological space. Show that $x_n\to_{n\to \infty} x$ if and only if $d(x_n,x)\to_{n\to \infty} 0.$ Hello, any idea for begin? Thanks.
  16. Mr Davis 97

    I Proving Convergence: Solving the Limit of 1/(6n^2+1) = 0

    I am trying to show that ##\displaystyle \lim \frac{1}{6n^2+1}=0##. First, we have to find an N such that, given an ##\epsilon > 0##, we have that ##\frac{1}{6n^2+1} < \epsilon##. But in finding such an N, I get the inequality ##n> \sqrt{\frac{1}{6}(\frac{1}{\epsilon}-1)}##. But clearly with...
  17. J

    A Newton's Generalized Binomial Theorem

    I'm trying to expand the following using Newton's Generalized Binomial Theorem. $$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$ where $$0<\delta<<1$$ But the condition for this formula is that...
  18. Another

    Testing Absolute Convergence of ∑(-2)n+1/n+5n

    ∑ (-2)n+1/n+5n Test this series Absolute Convergence ? ∑|an| = ∑(2)n+1/n+5n if the sum of |an| converges, than the sum of an converges ∑|an| = ∑(2)n+1/n+5n I can use Comparison Test? I can choose series bn = ∑ 2n/5n ?
  19. Leonardo Machado

    A Convergence of lattice Ising model

    Hello everyone. I'm working on a program to solve 2D Ising model of magnetic materials, using a system with 10x10 spins for simplicity at a temperature of 1E-8 K. I'm using this parameters to get a faster result of m=1 and guarantee it is correct. but... For now i already pass 300 Monte Carlo's...
  20. S

    I Hyperreal Convergence: Is It 0 or Infinitesimal?

    I have always thought that non-constant sequences that converge toward 0 in the reals converge toward an infinitesimal in the hyperreals, but recently I have questioned my presumption. If ##(a_n)\to0## in ##R##, wouldn't the same seuqnece converge to 0 in ##*R##? These two statements should...
  21. Mr Davis 97

    Finding Maclaurin expansion and interval of convergence

    Homework Statement Find the Maclaurin series and inverval of convergence for ##f(x) = \log (\cos x)## Homework EquationsThe Attempt at a Solution I used the fact that ##\log (\cos x) = \log (1+ (\cos x - 1))##, and the standard expansions for ##\cos x## and ##\log (x+1)## to get that...
  22. J

    MHB Testing Convergence, Calculus II

  23. karush

    MHB Find Power Series Representation for $g$: Interval of Convergence

    $\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$ $\textrm{ or Integrating the power series for $f$ perhaps more than once}$ \begin{align*}\displaystyle f(x)&=\frac{1}{1-3x} \\ &=\sum_{k=1}^{\infty} \end{align*} $\textsf{b. Give interval of convergence...
  24. S

    I Pointwise and Uniform Convergence

    Hello! Can someone explain to me in an intuitive way the difference between pointwise and uniform convergence of a series of complex functions ##f_n(z)##? Form what I understand, the difference is that when choosing an "N" such that for all ##n \ge N## something is less than ##\epsilon##, in the...
  25. M

    MHB Power series and uniform convergence.

    Hi. I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not. I hope someone can help me with this.
  26. evinda

    MHB Why do we have fast convergence?

    Hello! (Wave) Suppose that we have $u(x,t)= \frac{80}{\pi} \sum_{n=1,3,5, \dots}^{\infty} \frac{1}{n} e^{-\frac{n^2 \pi^2 a^2 t}{2500}} \sin{\frac{n \pi x}{50}}$. According to my notes, the negative exponential factor at each term of the series has as a result the fast convergence of the...
  27. M

    Region of convergence Z-transform

    Hello everyone. Iam just learning the z-transform for discrete signals and I can't get my head around the Region of covergence (ROC). As far as I have understood describes the ROC if the z-transform excists or not ? But how to I actually calculate it? Is there any kind of formula? I all...
  28. tl_ccc

    Nonlinear contact convergence problem in ANSYS Workbench?

    I am using the static structural module of ANSYS workbench to do a simulation. In my model, there is a gear and a spring which presses against the gear, moves along it and pushes it to turn counterclockwise. These two objects are in frictional contact. In my calculation, I always have the...
  29. A

    A 2D Finite Difference Convergence Rate Issues

    I have completed a 2D finite difference code in MATLAB that has a domain of (0,1)x(0,1) and has Dirichlet Boundary Conditions of value zero along the boundary. I get convergence rates of 2 for second order and 4 for fourth order. My issue now is that I'm now wanting to change the domain to a...
  30. A

    Convergence in distribution example

    Homework Statement Homework Equations [/B] Definition: A sequence X_1,X_2,\dots of real-valued random variables is said to converge in distribution to a random variable X if \lim_{n\rightarrow \infty}F_{n}(x)=F(x) for all x\in\mathbb{R} at which F is continuous. Here F_n, F are the...
  31. R

    Finding convergence of this series using Integral/Comparison

    Homework Statement series from n = 1 to infinity, (ne^(-n)) Homework EquationsThe Attempt at a Solution I want to use integral test. I know this function is: positive (on interval 1 to infinity) continous and finding derivative of f(x) = xe^(-x) I found it to be ultimately decreasing. So...
  32. C

    MHB Series Convergence: Ratio Test & Lim. n→∞

    I'm trying to determine if \sum_{n=1}^{\infty}\frac{{n}^{10}}{{2}^{n}} converges or diverges. I did the ratio test but I'm left with determining \lim_{{n}\to{\infty}}\frac{(n+1)^{10}}{2n^{10}} Any suggestions??
  33. K

    Complex Analysis/Radius of Convergence question.

    Homework Statement Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less. Homework Equations The Attempt at a Solution I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am...
  34. solour

    Why does (-1)^n(sin(pi/n)) converge when (sin(p/n)) diverges

    Homework Statement I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity. However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...
  35. karush

    MHB -z.54 find the radius of convergence

    $\tiny{10.7.37}$ $\displaystyle\sum_{n=1}^{\infty} \frac{6\cdot 12 \cdot 18 \cdots 6n}{n!} x^n$ find the radius of convergence I put 6 but that wasn't the answer
  36. M

    MHB No problem, happy to help! (Glad to hear it)

    Hey! :o I want to check which of the following sequences converges and from those that don't converge I want to check if it has a convergent subsequence. $\displaystyle{1, 1-\frac{1}{2}, 1, 1-\frac{1}{4}, 1, 1-\frac{1}{6}, \ldots}$ $\displaystyle{1, \frac{1}{2}, 1, \frac{1}{4}, 1...
  37. M

    MHB Converging Series: Tests & Tips for Finding Solutions

    Hi, I would like to as you you help please with finding whether the following three series converge. \sum_{1}^{\infty} (-1)kk3(5+k)-2k $$\sum_{k=1}^\infty(-1)^kk^3(5+k)^{-2k}$$ \sum_{2}^{\infty} sin(Pi/2+kPi)/(k0.5lnk) $$\sum_{k=2}^\infty\frac{\sin\left(\frac{\pi}{2}+k\pi\right)}{\sqrt k\ln...
  38. Vigardo

    Convergence problems in ANSYS batch mode but not in GUI

    Dear experts, I´m performing a non-linear buckling analysis under ANSYS Mechanical APDL (v14.5) using an input file that processes the last converged step to generate some etable output. When run in GUI everything goes fine: the non-linear buckling analysis is performed until it becomes...
  39. jlmccart03

    Series: Determine if they are convergent or divergent

    Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...
  40. binbagsss

    Elliptic functions proof -- convergence series on lattice

    Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
  41. M

    MHB How Do We Determine the Convergence of These Complex Integrals?

    Hey! :o I want to check the convergence of the following integrals: $\displaystyle{\int_2^{\infty}\frac{1}{x\left (\log (x)\right )^2}dx}$ We have that: \begin{equation*}\int_2^{\infty}\frac{1}{x\left (\log (x)\right )^2}dx=\lim_{b\rightarrow \infty}\int_2^b\frac{1}{x\left (\log (x)\right...
  42. I

    MHB What are the new formulas for x and y that will converge to $\sqrt{k}$?

    I'm not sure which category to post this question under :) I'm not sure if any of you are familiar with "Greek Ladders" I have these two formulas: ${x}_{n+1}={x}_{n}+{y}_{n}$ ${y}_{n+1}={x}_{n+1}+{x}_{n}$ x y $\frac{y}{x}$ 1 1 1 2 3 1.5 5 7 ~1.4 12 17 ~1.4 29 41...
  43. T

    Showing Convergent Subsequence Exists

    Homework Statement Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##. Homework Equations N/A...
  44. I

    I Convergence Vectors Calculus: Definition

    What is the definition of convergence in calculus for vectors?
  45. K

    Comparison test for series convergence (trig function)

    Homework Statement Use a comparison test to determine whether this series converges: \sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution At small values of x: \sin x\approx x a_{x}=\sin \frac{1}{x} b_{x}=\frac{1}{x} \lim...
  46. M

    MHB Pointwise and uniform convergence

    Hey! :o I want to check the pointwise and uniform convergence for the following sequences or series of functions: $f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=xe^{-nx}$ for all $n\in \mathbb{N}$ $f_n:[0, \infty)\rightarrow \mathbb{R}, f_n(x)=nxe^{-nx}$ for all $n\in \mathbb{N}$...
  47. Kernul

    Is the Summation Converging in the Given Interval?

    Homework Statement I'm give the following summation of functions and I have to see where it converge. $$\sum_{n = 1}^{\infty} \frac{(3 arcsin x)^n}{\pi^{n + 1}(\sqrt(n^2 + 1) + n^2 + 5)}$$ Homework EquationsThe Attempt at a Solution Putting ##3 arcsin x = y##, I already see that with the...
  48. Rectifier

    Is There a Constant Lower Bound for the Integral Test of Convergence?

    The problem I am trying to show that the following integral is convergent $$ \int^{\infty}_{2} \frac{1}{\sqrt{x^3-1}} \ dx $$The attempt ## x^3 - 1 \approx x^3 ## for ##x \rightarrow \infty##. Since ## x^3 -1 < x^3 ## there is this relation: ##\frac{1}{\sqrt{x^3-1}} > \frac{1}{\sqrt{x^3}}##...
  49. B

    Convergence of a Sequence in a Finer Topology

    Homework Statement Clearly if a sequence of points ##\{x_n\}## in some space ##X## with some topology, then the sequence will also converge when ##X## is endowed with any coarser topology. I suspect this doesn't hold for endowment of ##X## with a finer topology, since a finer topology amounts...
  50. I

    Convergence of sequence in metric space proof

    Homework Statement Let ##E \subseteq M##, where ##M## is a metric space. Show that ##p\in \overline E = E\cup E' \Longleftrightarrow## there exists a sequence ##(p_n)## in ##E## that converges to ##p##. ##E'## is the set of limit points to ##E## and hence ##\overline E## is the closure of...
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