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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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: "...
  12. 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...
  13. 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...
  14. D

    The convergence of a series

    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)##...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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}...
  20. 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)!}##...
  21. 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)...
  22. M

    MHB Check convergence of sequences

    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...
  23. 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...
  24. 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...
  25. 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...
  26. 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}}...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. M

    I Why is there a contradiction?

    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...
  32. mertcan

    Taylor Series Convergence

    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...
  33. 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 :/
  34. 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...
  35. 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##...
  36. 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.
  37. 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.
  38. 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...
  39. 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...
  40. A

    What is the radius of convergence for a series with logarithmic terms?

    Homework Statement This is from a complex analysis course: Find radius of convergence of $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ Homework Equations I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}## The Attempt at a Solution My first reaction is that this sum looks...
  41. JD_PM

    Analysis of an absolutely convergence of series

    Homework Statement - Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. - Suppose ##(y_n)_n## is...
  42. A

    I Unable to show the radius of convergence of a numeric series

    Hi, I've computed 512 terms of a power series numerically. Below are the first 20 terms. $$ \begin{align*} w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...
  43. M

    I Error estimate for iterative convergence

    On the following page on wikipedia: https://en.wikipedia.org/wiki/Fixed-point_iteration the section "Examples" has a second bullet point, where the author suggests ##q=0.85##, but how did they get this number? I tried googling everything and could not find out how ##q## is determined.
  44. J

    MHB Real Analysis - Convergence to Essential Supremum

    Problem: Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that $lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...
  45. M

    On the convergence of the three body disturbing function

    For the three-body-disturbing function expanded in multipolar orders with respect to the ratio of the semi major axes, the function converges for small ratios, how to check the convergence for a certain set of parameters? I'm using the work of the Laskar in his paper "Explicit expansion of the...
  46. F

    I Convergence of 2 sample means with 95% confidence

    I tried to derive an equation for one sample mean to converge to another sample mean within a 95% confidence interval, but I know I am wrong. Can someone tell me what I did wrong, and what is the correct formula? Suppose: ##\hat{x_1},\hat{\sigma_1},N## are a sample mean, standard deviation...
  47. K

    Absolute convergence of series

    Homework Statement Hello, I need some feedback on whether this reasons is correct. consider the series Examine the series for absolute convergence. Homework EquationsThe Attempt at a Solution How I have solved this, using the limit comparison test: we have: introducing we have that...
  48. Math Amateur

    MHB Pointwise Convergence .... Abbott, Example 6.2.2 (iii) .... ....

    I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ... I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on pointwise convergence... I need some help to understand the 'mechanics' of Example 6.2.2 (iii) ... Example 6.2.2 reads as...
  49. bhobba

    I What is convergence and 1+2+3+4....... = -1/12

    Ok - anyone that has done basic analysis knows the definition of convergence. The series 1-2+3-4+5... is for example obviously divergent (alternating series test). But wait a minute let's try something tricky and perform a transform on it, (its Borel summation, but that is not really relevant...
  50. Jazzyrohan

    I Convergence of an infinite series

    For a series to be convergent,it must have a finite sum,i.e.,limiting value of sum.As the sum of n terms approaches a limit,it means that the nth term is getting smaller and tending to 0,but why is not the converse true?Should not the sum approach a finite value if the nth term of the series is...
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