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. 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...
  2. 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(...
  3. 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...
  4. 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!
  5. 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) >...
  6. 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!
  7. 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
  8. 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!
  9. 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...
  10. M

    MHB Do These Integrals Converge or Diverge?

    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...
  11. 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...
  12. 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...
  13. 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...
  14. M

    MHB What conditions guarantee convergence of Newton's method for approximating pi/2?

    Hey! 😊 I have calculated an approximation to $\frac{\pi}{2}$ using Newton's method on $f(x)=\cos (x)$ with starting value $1$. After 2 iterations we get $1,5707$. Which conditions does the starting point has to satisfy so that the convergence of the sequence of the Newton iterations to...
  15. P

    Coulomb's Law and Conditional Convergent Alternating Harmonic Series

    Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...
  16. S

    I Convergence of sequences of functions with differing domains?

    Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...
  17. M

    MHB Sequence of functions : pointwise & uniform convergence

    Hey! 😊 Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$. For an integer $m>a$ we have that \begin{equation*}0 \leq...
  18. M

    MHB Checking Convergence of Series: Inequalities & Tests

    Hey! 😊 I want to check the convergence for the below series. - $\displaystyle{\sum_{n=1}^{+\infty}\frac{\left (n!\right )^2}{\left (2n+1\right )!}4^n}$ Let $\displaystyle{a_n=\frac{\left (n!\right )^2}{\left (2n+1\right )!}\cdot 4^n}$. Then we have that \begin{align*}a_{n+1}&=\frac{\left...
  19. R

    Engineering Relationship between the solution convergence and boundary conditions

    I create an algorithm that can solve [K]{u}={F} for atomic structure, but the results are not converge Do the boundary conditions affect the convergence of the resolution of a system of nonlinear partial equations? And how to know if the solution is diverged because of the boundary conditions...
  20. Coltrane8

    I Spherical Harmonics Expansion convergence

    In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...
  21. Math Amateur

    I Convergence .... Singh, Example 4.1.1 .... .... Another Question ....

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ... I need some further help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows: In the above example from Singh we read the...
  22. Math Amateur

    I Convergence in Topological Spaces .... Singh, Example 4.1.1 .... ....

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ... I need help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows: In the above example from Singh we read the following: "...
  23. F

    Possible Values of x for Convergence of Power Series

    We transform the series into a power series by a change of variable: y = √(x2+1) We have the following after substituting: ∑(2nyn/(3n+n3)) We use the ratio test: ρn = |(2n+1yn+1/(3n+1+(n+1)3)/(2nyn/(3n+n3)| = |(3n+n3)2y/(3n+1+(n+1)3)| ρ = |(3∞+∞3)2y/(3∞+1+(∞+1)3)| = |2y| |2y| < 1 |y| = 1/2...
  24. F

    Find the interval of convergence of this power series

    ∑((√(x2+1))n22/(3n+n3)) We use the ratio test: ρn = |2(3n+n3)√(x2+1)/(3n+1+(n+1)3)| ρ = |2√(x2+1)| ρ < 1 |2√(x2+1)| < 1 No "x" satisfies this expression, so I conclude the series doesn't converge for any "x". However the answer in the book says the series converges for |x| < √(5)/2. What am...
  25. D

    Does This Series Converge for Different Values of Alpha?

    The series ##\sum_{n=0}^\infty \left( ne^{\frac 3 n}-n \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## i did ##\sum_{n=0}^\infty n\left( e^{\frac 3 n}-1 \right) \left ( \sin \frac {\alpha} {n} - \frac 5 n\right)## for n going to infinity ## \left( e^{\frac 3 n}-1 \right)##...
  26. F

    Find the Interval of Convergence of this Power Series: ∑(x^2n/(2^nn^2))

    ∑(x2n/(2nn2)) We use the ratio test: ρn = |(x2n2/(2(n+1)2)| ρ = |x2/2| ρ < 1 |x2| < 2 |x| = √(2) We investigate the endpoints: x = 2: ∑(4n/(2nn2) = ∑(2n/n2)) We use the preliminary test: limn→∞ 2n/n2 = ∞ Since the numerator is greater than the denominator. Therefore, x = 2 shouldn't be...
  27. D

    Verify the convergence or divergence of a power series

    At the exam i had this power series but couldn't solve it ##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k## i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...
  28. D

    Power series: radius of convergence

    ##\sum_{k=0}^\infty \frac {2^n+3^n}{4^n+5^n} x^n## in order to find the radius of convergence i apply the root test, that is ##\lim_{n \rightarrow +\infty} \sqrt [n]\frac {2^n+3^n}{4^n+5^n}## ##\lim_{n \rightarrow +\infty} \left(\frac {2^n+3^n}{4^n+5^n}\right)^\left(\frac 1 n\right)=\lim_{n...
  29. D

    Set of convergence of a Power series

    given the following ##\sum_{n=0}^\infty n^2 x^n## in order to find the radius of convergence i do as follows ##\lim_{n \rightarrow +\infty} \left |\sqrt [n]{n^2}\right|=1## hence the radius of convergence is R=##\frac 1 1=1## |x|<1 Now i have to verify how the series behaves at the...
  30. D

    Study the convergence and absolute convergence of the following series

    ## \sum_{n=1}^\infty (-1)^n \frac {log(n)}{e^n}## i take the absolute value and consider just ## \frac {log(n)}{e^n}## i check by computing the limit if the necessary condition for convergence is satisfied ##\lim_{n \rightarrow +\infty} \frac {log(n)}{e^n} =\lim_{n \rightarrow +\infty}...
  31. D

    Checking the convergence of this numerical series using the ratio test

    ## \sum_{n=0}^\infty \frac {(2n)!}{(n!)^2} ## ##\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}## that becomes ##\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}## ##\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}##...
  32. benorin

    I Convergence of a sequence of sets

    I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please. I wish to follow up on this thread with a definition of convergence of a sequence of sets from Baby Rudin (Principles of Mathematical Analysis, 3rd ed., Rudin) pgs. 304-305: (pg. 304)...
  33. M

    MHB Is the Convergence of These Sequences Correctly Determined?

    Hey! :o Check the below sequences for convergence and determine the limit if they exist. Justify the answer. $\displaystyle{f_n:=\left (1-\frac{1}{2n}\right )^{3n+1}}$ $\displaystyle{g_n:=(-1)^n+\frac{\sin n}{n}}$ I have done the following: $\displaystyle{f_n:=\left...
  34. J

    Why does the Euler approximation fail for the Airy or Stokes equation?

    I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
  35. J

    MHB Weak Convergence to Normal Distribution

    Problem: Let $X_n$ be independent random variables such that $X_1 = 1$, and for $n \geq 2$, $P(X_n=n)=n^{-2}$ and $P(X_n=1)=P(X_n=0)=\frac{1}{2}(1-n^{-2})$. Show $(1/\sqrt{n})(\sum_{m=1}^{n}X_n-n/2)$ converges weakly to a normal distribution as $n \rightarrow \infty$.Thoughts: My professor...
  36. 0

    Integral of 1/ln(x). Convergence test

    Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance. The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges. ##\int_1^\infty (\ln(x))^n dx## If n = 0, I...
  37. karush

    MHB 11.8.4 Find the radius of convergence and interval of convergence

    Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (1) $$a_n=\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (2) $$\left|\dfrac{a_{a+1}}{a_n}\right| =\left|\dfrac{(-1)^{n+1} x^{n+1}}{\sqrt[3]{n+1}}...
  38. Math Amateur

    MHB Absolute and Conditional Convergence .... Sohrab Proposition 2.3.22 ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Proposition 2.3.22 ... Proposition 2.3.12 reads as follows: Can someone please demonstrate (formally and...
  39. Math Amateur

    MHB Convergence of Geometric Series .... Sohrab, Proposition 2.3.8 .... ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with an aspect of the proof of Proposition 2.3.8 ... Proposition 2.3.8 and its proof read as follows: In the above proof by...
  40. S

    True or false question regarding the convergence of a series

    I think ##\lim_{n\rightarrow \infty} a_n = 0## since by direct substitution the value of limit won't be equal to 2 so by direct substitution we must get indeterminate form. Then how to check for ##\sum_{n=1}^\infty a_n##? I don't think divergence test, integral test, comparison test, limit...
  41. M

    Uniform convergence of a sine series

    I'm not too sure how to use the hint here. What I had so far was this: an odd extension of ##f## implies ##f = \sum_{k=1}^\infty b_k \sin(k x)##. Notice for ##m>n## $$ \left|\sum_{k=1}^m b_k\sin(k x) - \sum_{k=1}^n b_k\sin(k x)\right| = \left| \sum_{k=n+1}^m b_k\sin(k x)\right| \leq...
  42. M

    I Why Does the Electric Field Calculation Diverge Inside the Volume?

    Let: ##\nabla## denote dell operator with respect to field coordinate (origin) ##\nabla'## denote dell operator with respect to source coordinates The electric field at origin due to an electric dipole distribution in volume ##V## having boundary ##S## is: \begin{align} \int_V...
  43. mertcan

    How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?

    Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a...
  44. D

    MHB Radius and Interval of Convergence for (x/sin(n))^n

    Find Radius and Interval of Convergence for \sum_{1}^{\infty}(\frac{x}{sinn})^{n}. I don`t have any ideas how to do that :/
  45. F

    A Consistency Versus Convergence, seeking intuition

    What is the definition of consistency? I have seen a proof that shows a finite difference scheme is consistent, where they basically plug a true solution ##𝑢(𝑡)## into a finite difference scheme, and they get every term, for example ##𝑢^{𝑖+1}_𝑗## and ##𝑢^𝑖_{𝑗+1}##, using taylors polynomials...
  46. E

    A How to get a converging solution for a second order PDE?

    I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where ##G_{zy}##, ##G_{zx}##, ##\theta##...
  47. J

    MHB Power Series for f(x) and Radius of Convergence

    f(x) = 4x/(x-3)^2 Find the first five non-zero terms of power series representation centered at x = 0. Also find the radius of convergence.
  48. Entertainment Unit

    Test the following series for convergence or divergence

    Homework Statement Test the following series for convergence or divergence. ##\sum_{n = 1}^{\infty} \frac {\sqrt n} {e^\sqrt n}## Homework Equations None that I'm aware of. The Attempt at a Solution I know I can use the Integral Test for this, but I was hoping for a simpler way.
  49. Entertainment Unit

    Convergence of a series given in non-closed form

    Homework Statement Determine whether the given series is absolutely convergent, conditionally convergent, or divergent. ##\frac{1}{3} + \frac{1 \cdot 4}{3 \cdot 5} + \frac{1 \cdot 4 \cdot 7}{3 \cdot 5 \cdot 7} + \frac{1 \cdot 4 \cdot 7 \cdot 10}{3 \cdot 5 \cdot 7 \cdot 9} + \ldots + \frac{1...
  50. opus

    Intervals of Convergence- Power Series

    Homework Statement Hello. I'm not entirely sure what this question is asking me, so I'll post it and let you know my thoughts, and any input is greatly appreciated. If the series ##\sum_{n=0}^\infty a_n(x-4)^n## converges at x=6, determine if each of the intervals shown below is a possible...
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