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. 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...
  2. 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##.
  3. 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...
  4. 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.
  5. 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...
  6. 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...
  7. 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 ?
  8. 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...
  9. 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...
  10. 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...
  11. J

    MHB Testing Convergence, Calculus II

  12. 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...
  13. 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...
  14. 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.
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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??
  22. 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...
  23. 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...
  24. 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
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. I

    I Convergence Vectors Calculus: Definition

    What is the definition of convergence in calculus for vectors?
  34. 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...
  35. 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 $n\in \mathbb{N}$...
  36. 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...
  37. 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}}##...
  38. 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...
  39. 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...
  40. MAGNIBORO

    I What do I need to know to understand Uniform convergence?

    Hi, I started to study the function of Weierstrass (https://en.wikipedia.org/wiki/Weierstrass_function) And in one part says that the sum of continuous functions is a continuous function. i understand this but the Limiting case is a different history depend of the convergence, so what i need...
  41. C

    I Convergence of Taylor series in a point implies analyticity

    Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
  42. B

    Radius of Convergence for Ratio Test in Calculus Questions

    Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...
  43. M

    Summation of sin(pi*n/2)/2: Is the Execution Correct?

    Homework Statement \sum_{n=2}^{\infty}sin(\frac{\pi*n}{2})/{2}I don't have a solution, and wondered if the execution is correct. The Attempt at a Solution I thought that one can use comparison test where; \sum b_n= \frac{1}{n^{1/2}}. Since p<1 ---> divergent. But many of the students says it...
  44. M

    MHB Finding Radius of Convergence for Series: n/2^n and 1/(4+(-1)^n)^3n

    Hey! :o I want to find for the following series the radius of convergence and the set of $x\in \mathbb{R}$ in which the series converges. $\displaystyle{\sum_{n=0}^{\infty}\frac{n}{2^n}x^{n^2}}$ $\displaystyle{\sum_{n=0}^{\infty}\frac{1}{(4+(-1)^n)^{3n}}(x-1)^{3n}}$ I have done the...
  45. Kernul

    Punctual and uniform convergence

    Homework Statement ##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##. We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if $$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x...
  46. T

    MHB Series Convergence Or Divergence

    I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...
  47. P

    I Is the convergence of an infinite series mere convention?

    It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality. For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1, since there would always be an infinitely small fraction that made the second half unreachable relative to the...
  48. sumner

    A Convergence of an infinite series of exponentials

    I have a set of data that I've been working with that seems to be defined by the sum of a set of exponential functions of the form (1-e^{\frac{-t}{\tau}}). I've come up with the following series which is the product of a decay function and an exponential with an increasing time constant. If this...
  49. NihalRi

    Finding the interval of convergence

    Homework Statement The question was to find the interval of convergence for a series. Homework Equations an+1/an The Attempt at a Solution
  50. Battlemage!

    B Absolute convergence Text Book question: Boas 3rd Ed

    In Mary L. Boas' Mathematical Methods in the Physical Science, 3rd ed, on page 17 it goes over absolute convergence, and defines the test for alternating series as follows: An alternating series converges if the absolute value of the terms decreases steadily to zero, that is, if |an+1| ≤ |an|...
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