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. S

    I "Magic" regulating functions for divergent series

    This recent video on Numberphile revisits -1 / 12 😱 after a hiatus of nearly 10 years. One point that they make is that there are infinitely many choices of regulating function that converge directly to the correct value (e.g. -1/12) without having to throw away "infinities" or terms of order...
  2. L

    Checking series for convergence

    Hi, I am having problems with task d) I now wanted to check the convergence using the quotient test, so ## \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}| < 1## I have now proceeded as follows: ##\frac{a_{n+1}}{a_n}=\frac{\Bigl( 1 + \frac{1}{k+1} \Bigr)^{(k+1)^2}}{3^{k+1}} \cdot...
  3. P

    Show Picard iteration diverges

    For an example of a Picard iteration, see here. In this case, we have \begin{align} &x_0(t)=x(0)=0,\nonumber\\ &x_1(t)=x_0(t)+\int_0^t \big(1+(x_0(s)-s)\big)^2ds=t+\frac{t^3}{3},\nonumber \\ &x_2(t)=x_0(t)+\int_0^t \big(1+(x_1(s)-s)\big)^2ds=t+\frac{t^7}{3^27},\nonumber\\ &\cdots \nonumber...
  4. person123

    I Make Intelligent Initial Guesses for Newton-Raphson Programmatically

    I'm writing code to numerically solve a single variable equation, currently with Newton Raphson's method. Right now, I'm just using an initial guess of 1, and reporting a failure if it doesn't converge. While it usually works, it does of course fails for many functions with asymptotes or other...
  5. Euge

    POTW A Test for Absolute Convergence of a Series

    Let ##\{a_n\}_{n = 1}^\infty## be a sequence of real numbers such that for some real number ##p > 1##, ##\frac{a_n}{a_{n+1}} = 1 + \frac{p}{n} + b_n## where ##\sum b_n## converges absolutely. Show that ##\sum a_n## also converges absolutely.
  6. Euge

    POTW Convergence of Random Variables in L1

    Let ##\{X_n\}## be a sequence of integrable, real random variables on a probability space ##(\Omega, \mathscr{F}, \mathbb{P})## that converges in probability to an integrable random variable ##X## on ##\Omega##. Suppose ##\mathbb{E}(\sqrt{1 + X_n^2}) \to \mathbb{E}(\sqrt{1 + X^2})## as ##n\to...
  7. W

    Poor convergence using Fe HMOs for FMO in Gamess

    Hello! I'm trying to model an interaction between ligands and a heme group using FMO in Gamess. I've tried to make HMOs for the FMOBND section of the input file (using an Fe-F complex), the HMOs are shown below: STO-3G 19 5 0 1 0.992624, 0.019366, 0.000000, 0.000000,-0.000001,-0.014838...
  8. H

    For what values of ##\beta## does the series converge?

    For what values of β the following series converges $$ \sum_{k=1}^{\infty} k^{\beta} \left( \frac{1}{\sqrt k} - \frac{1}{\sqrt {k+1}}\right) $$ I thought of doing it like this $$ \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta} }{\sqrt{k+1}}$$ $$0 \lt \frac{k^{\beta} }{\sqrt k} - \frac{k^{\beta}...
  9. chwala

    Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x

    ##a_n= \left[\dfrac {\ln (n)^2}{n}\right]## We may consider a function of a real variable. This is my approach; ##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]## Applying L'Hopital's rule we shall have; ##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...
  10. Euge

    POTW Convergence in Probability

    Prove that if ##\{X_n\}_{n = 1}^\infty## is a sequence of real random variables on probability space ##(\Omega, \mathscr{F},\mathbb{P})## such that ##\lim_n \mathbb{E}[X_n] = \mu## and ##\lim_n \operatorname{Var}[X_n] = 0##, then ##X_n## converges to ##\mu## in probability.
  11. A

    Convergence of a Series: Radius and Endpoints

    Greetings According to my understanding: if x converges in 4 means that the series converges -1<x+3<7 but the solution says C Any hint? thank you!
  12. C

    A Convergence issue in this Least Squares calculation

    I'm computing the trajectory of a moving body and my net is composed by 5 stations. My observations are DTOA: difference in time of Arrival (they have been linearized). I am trying to use Least Squares with a linear model: Y = Ax + b, where Y are the observed measurements (DTOA), A the design...
  13. H

    Why are there clouds over the Intertropical Convergence Zone?

    Intertropical Tropical Zone is the zone where north-east and south-east trade winds converge. This zone usually occurs over (I don’t know if “on” should be here) the equator. In the book The Atmosphere: An Introduction to Meteorology by Lutgens and Tarbuck (13th Edition), Figure 7.9 reads I...
  14. C

    I Proving a convergent sequence is bounded

    Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...
  15. F

    A Residual of PDEs as convergence criteria of numerical solution

    Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution...
  16. A

    The convergence Criteria ratio

    Greetings all I have a question regarding the convergence criteria ratio, abs(an+1/an) or the n√abs(an) when the limit tend to a value less than 1 does it mean the serie is convergent or absolutely convergent? Thank you!
  17. A

    Question about the convergence of a series

    Greetings! I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct? thank you! when we
  18. chwala

    Bounded and monotonic sequences - Convergence

    I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...
  19. A

    The radius of convergence of a series

    Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!
  20. H

    Prove that the inner product converges

    I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...
  21. L

    I Convergence of this Laplace transformation

    I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is: X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ). Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s). If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...
  22. carbondio379

    Is Convergence Possible from A to C?

    If there are points A and B in space, If an object travels a distance from A to any other point between A and B, does that count as converging? And if the object from point A reach point C (a point between A and B) without traveling a distance, does that count as converging too? Like if the...
  23. M

    MHB Is the Convergence Uniform?

    Hey! :giggle: We have the sequence of functions $$f_n=\sin (x)-\frac{nx}{1+n^2}$$ I want to check the pointwise andthe uniform convergence. We have that $$f^{\star}(x)=\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}\left (\sin (x)-\frac{nx}{1+n^2}\right )=\sin(x)$$ So $f_n(x)$...
  24. M

    MHB Extrema and convergence of sequence

    Hey! :giggle: For $n\in \mathbb{N}$ let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ given by $f_n(x)=\frac{x+2n}{x^2+n}$. (a) Determine all (local and global) extrema of $f_n$ and the behaviour for $|x|\rightarrow \infty$. Make a sketch for $f_n$ and $f_n'$. Show that there exists $x_1<x_2<x_3<x_4$...
  25. A

    The convergence of a numerical series

    Greetings here is the exercice My solution was as n^2+n+1/(n+1) tends asymptotically to n then the entire stuffs inside the sinus function tends to npi which make it asymptotically equal to sin(npi) which is equal to 0 and consequently the sequence is Absolutely convergent Here is the...
  26. A

    Finding the Radius of Convergence for Y=6x+16 - Troubleshooting and Solution

    Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...
  27. Eclair_de_XII

    B Convergence of a sequence of averages of a convergent sequence

    Let ##\epsilon>0##. Then there is an integer ##N>0## with the property that for any integer ##n\geq N##, ##|a_n-A|<\epsilon##, where ##A\in\mathbb{R}##. If for all positive integers ##n##, it is the case that ##|a_n-A|<\epsilon##, then the following must hold: \begin{eqnarray}...
  28. W

    Series Convergence: What Can the Nth Term Test Tell Us?

    I'm not sure which test is the best to use, so I just start with a divergence test ##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}## The +3 and +1 are negligible ##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}## So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test...
  29. yucheng

    Purcell EM Problem 1.2: Theory Behind Numerical Solution?

    The author start of with $\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$ and arrives at the equation $y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this...
  30. E

    I Fredholm's alternative & L2 convergence

    Hello everyone, I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem. Assume that we are...
  31. A

    I Understanding the Laplace Transform of cos(t)/t

    So, I know the direct definition of the Laplace Transform: $$ \mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...
  32. Fochina

    Finding the convergence of a parametric series

    It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
  33. A

    A problem with the convergence of a series

    Good day I have a question about the convergence of the following serie I understand that the racine test shows that it an goes to 2/3 which makes it convergent but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim...
  34. M

    MHB Uniform convergence - Length of graph

    Hey! :giggle: We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left...
  35. M

    MHB Convergence as for the cofinite topology on R

    Hey! :giggle: Does the sequence $x_n=\frac{1}{n}$ converges as for the cofinite topology on $\mathbb{R}$ ? If it converges,where does it converge? Could you explain to me what exactly is meant by "cofinite topology on $\mathbb{R}$" ? Do we have to define first this set and then check if we...
  36. M

    MHB How can we prove the convergence of recursive defined sequences?

    Hey! :giggle: a) Check the convergence of the sequence $a_n=\left (\frac{n+2000}{n-2000}\right)^n$, $n>1$. If it converges calculate the limit. b) Check the convergence of the recursive defined sequence $a_n=\frac{a_{n-1}}{a_{n-1}+2}$, $n>1$, with $a_1=1$.For a) we have $$a_n=\left...
  37. L

    MHB Fixed point iteration convergence

    Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $ p_{0} \in \left [ 0,1 \right ]$? Justify your answer. a.$ g(x) = e^{\frac{-x}{2}}$ b.$ g(x) = 3x - 1$ Let me attempt for part a first...
  38. JD_PM

    Checking convergence of Gaussian integrals

    a) First off, I computed the integral \begin{align*} Z(\lambda) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}-\frac{\lambda}{4!}x^4\right) \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}\right) \exp\left(...
  39. K

    I Convergence criterion for Newton-Raphson

    The Newton-Raphson algorithm is well-known: ##x_{n+1} = x_n - \frac{f(x_n)}{f'(x_{n})}## Looking at a few implementations online, I have encountered two methods for convergence: 1) The first method uses the function value of the last estimate itself, ##f(x_n)## or ##f(x_{n+1})##. Since at...
  40. A

    Problem in finding the radius of convergence of a series

    Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!
  41. A

    Studying the convergence of a series with an arctangent of a partial sum

    Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...
  42. A

    Convergence of a series involving ln() terms in the denominator of a fraction

    good day I want to study the convergence of this serie and want to check my approch I want to procede by asymptotic comparison artgln n ≈pi/2 n+n ln^2 n ≈n ln^2 n and we know that 1/(n ln^2 n ) converge so the initial serie converge many thanks in advance!
  43. A

    Discussing the Convergence of a Series: Get My Opinion!

    Good day I want to study the connvergence of this serie I already have the solution but I want to discuss my approach and get your opinion about it it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge? many thanks in advance
  44. A

    Problem with series convergence — Taylor expansion of exponential

    Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!
  45. A

    Problem studying the convergence of a series

    Good day here is the exercice and here is the solution that I understand very well but I have a confusion I hope someone can explain me if I take the taylor expansion of sin ((n^2+n+1/(n+1))*pi)≈n^2+n+1/(n+1))*pi≈n*pi which diverge! I know something is wrong in my logic please help me many...
  46. M

    MHB Check convergence of integrals

    Hey! :giggle: I want to check if the following integrals converge or diverge. 1 . $\displaystyle{\int_0^{+\infty}t^2e^{-t^2}\, dt}$ 2. $\displaystyle{\int_e^{+\infty}\frac{1}{t^n\ln t}\, dt, \ n\in \{1,2\}}$ 3. $\displaystyle{\int_0^{+\infty}\frac{\sin t}{\sqrt{t}}\, dt}$ 4...
  47. D

    I Why are analyticity and convergence related in complex analysis?

    Hello, I am currently reading about the Residue Theorem in complex analysis. As a part of the proof, Mary Boas' text states how the a_n series of the Laurent Series is zero by Cauchy's Theorem, since this part of the Series is analytic. This appears to then be related to convergence of the...
  48. DuckAmuck

    I Einstein Field Eqns: East/West Coast Metrics

    My questions is: Depending on which metric you choose "east coast" or "west coast", do you have to also mind the sign on the cosmological constant in the Einstein field equations? R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \pm \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} For example, if you...
  49. M

    MHB Monotonically convergence to the root

    Hey! 😊 We have the following iteration from Newton's method \begin{align*}x_{k+1}&=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\frac{x_k^n-a}{nx_k^{n-1}}=\frac{x_k\cdot nx_k^{n-1}-\left (x_k^n-a\right )}{nx_k^{n-1}}=\frac{ nx_k^{n}-x_k^n+a}{nx_k^{n-1}}\\ & =\frac{ (n-1)x_k^{n}+a}{nx_k^{n-1}}\end{align*} I...
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