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. N

    Need to find if a sequence of functions has uniform convergence

    Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...
  2. B

    Proving Convergence in C[0,1] with Integral Norm

    Homework Statement Show that a sequence ##f_n \to f \in C[0,1]## with the sup norm ##|| ||_\infty##, then ##f_n \to f \in C[0,1]## with the integral norm. The Attempt at a Solution given ##\epsilon > 0 \exists n_0 \in N## s.t ##||(fn-f) (x)|| < \epsilon \forall n > n_0## with ##...
  3. A

    Showing X Vector Concentrated Around Unit Sphere Convergence of Mean-Squared

    Homework Statement Let Y_i be standard normal random variables, and let X be an N vector of random variables, X=(X_1, ..., X_N) where X_i = 1/{sqrt{N}} * Y_i. I want to show that as N goes to infinity, the vector X becomes "close" to the unit sphere. Homework Equations The...
  4. S

    Trying to find the radius of convergence of this complicated infinite series

    Homework Statement k is a positive integer. \sum^{\infty}_{n=0} \frac{(n!)^{k+2}*x^{n}}{((k+2)n)!} Homework Equations The Attempt at a Solution I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get...
  5. S

    Trying to find the interval of convergence of this series: run into a problem

    Homework Statement The series: \sum^{n=\infty}_{n=0} \frac{(-1)^{n+2}(x^{3}+8)^{n+1}}{n+1} Homework Equations The Attempt at a Solution using the ratio test, I get the following: |x^{3}+8|<1, but I know that the radius of convergence must be in the form: |x-a|<b, where...
  6. S

    Trying to find interval of convergence for a geometric series

    Homework Statement here is the series: \sum^{\infty}_{n=0}x(-15(x^{2}))^{n} Homework Equations The Attempt at a Solution I know that -1<-15x^{2}<1 for convergence (because of geometric series properties) but I run into a problem here: -1/15<x^{2}<1/15 You can't...
  7. D

    Convergence and stability in multivariate fixed point iteration

    Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...
  8. H

    Which Method to Use for Testing Convergence in Integrals with Substitution?

    Homework Statement Use integration, the direct comparison test, or the limit comparison test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. Homework Equations ∫sinθdθ/√π-) The Attempt at a Solution I don't know which method...
  9. J

    Extending radius of convergence by analytic continuation

    Hi, Suppose I have an analytic function f(z)=\sum_{n=0}^{\infty} a_n z^n the series of which I know converges in at least |z|<R_1, and I have another function g(z) which is analytically continuous with f(z) in |z|<R_2 with R_2>R_1 and the nearest singular point of g(z) is on the circle...
  10. Δ

    Uniform convergence of a quotient

    Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...
  11. B

    MHB Understanding Normed Linear Spaces: Convergence in C[0,1]

    Folks, I am looking at this task. 1) What does it mean to say a sequence converges in a normed linear space? 2) Show that if a sequence fn converges to f in C[0,1] with sup norm then it also converges with the integral norm? Any idea on how I tackle these? thanks
  12. S

    Comparison test for convergence problem: why is this incorrect?

    Homework Statement The original question is posted on my online-assignment. It asks the following: Determine whether the following series converges or diverges: \sum^{\infty}_{n=1}\frac{3^{n}}{3+7^{n}} There are 3 entry fields for this question. One right next to the series above...
  13. C

    What is the Convergence Criterion for a Bounded Sequence with a Common Limit?

    Homework Statement Assume a_n is a bounded sequence with the property that every convergent sub sequence of a_n converges to the same limit a. Show that a_n must converge to a. The Attempt at a Solution Could I do a proof by contradiction. And assume that a_n does not converge...
  14. D

    Test for convergence of the series

    Homework Statement Q) Summation from 1 to infinity (1+(-1)^i) / (8i+2^i) This series apparently converges and I can't figure out why. Homework Equations The Attempt at a Solution (1+(-1)^i) / i(8+2^i/i) Taking the absolute value of the above generalization...
  15. J

    Proving the Minimum Radius of Convergence for a Sum of Taylor Series

    Homework Statement how to prove that radius of convergence of a sum of two series is greater or equal to the minimum of their individual radii i don't know how to begin, can someone give me some ideas?
  16. P

    MHB Testing Uniform Convergence of Complex Function Sequences with Natural Numbers

    How do I determine whether the following sequences of complex functions converge uniformly? i) z/n ii)1/nz iii)nz^2/(z+3in) where n is natural number
  17. L

    Convergence of a sequence of integrals

    Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...
  18. alexmahone

    MHB Analysis of Convergence for Series a) and b)

    Do the following series converge or diverge? a) $\displaystyle \sum\frac{1}{n!}\left(\frac{n}{e}\right)^n$ b) $\displaystyle\sum\frac{(-1)^n}{n!}\left(\frac{n}{e}\right)^n$ My attempt: a) $\displaystyle\frac{1}{n!}\left(\frac{n}{e}\right)^n\approx\frac{1}{\sqrt{2\pi n}}$ (Stirling’s formula)...
  19. C

    Does pointwise convergence imply convergence of integral on C[0,1]?

    Homework Statement Does pointwise convergence imply convergence of integral on C[0,1]? Homework Equations The Attempt at a Solution Would like to verify that f(t)=t^n is a counter example of this. Pointwise convergence goes to 0 on [0,1) and 1 on [1]. Norm(1) is unbounded. I...
  20. alexmahone

    MHB Is the Series $\displaystyle\sum\frac{\sin n}{n}$ Absolutely Convergent?

    Using the fact that the interval $\displaystyle\left[2k\pi+\frac{\pi}{4},\ 2k\pi+\frac{3\pi}{4}\right]$ contains an integer, prove that $\displaystyle\sum\frac{\sin n}{n}$ is not absolutely convergent.
  21. A

    Finding the Radius and Interval of Convergence of a Series

    Homework Statement Find the radius of convergence and interval of convergence of the series: as n=1 to infinity: (n(x-4)^n) / (n^3 + 1) Homework Equations convergence tests The Attempt at a Solution i tried the ratio test but i ended up getting x had to be less than 25/4 ...
  22. C

    Is the Sequence x_{n+1}=\sqrt{2x_n} Converging?

    Homework Statement Show that \sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}} converges and find the limit. The Attempt at a Solution I can write it also like this correct 2^{\frac{1}{2}},2^{\frac{1}{2}}2^{\frac{1}{4}},2^{\frac{1}{2}}2^{\frac{1}{4}}2^{\frac{1}{8}} so each time i...
  23. alexmahone

    MHB Absolute Convergence: Proving $\sum a_n$ is Convergent

    Prove that if $\displaystyle \left|\frac{a_{n+1}}{a_n}\right|\le\left|\frac{b_{n+1}}{b_n}\right|$ for $\displaystyle n\gg 1$, and $\displaystyle\sum b_n$ is absolutely convergent, then $\displaystyle\sum a_n$ is absolutely convergent.
  24. alexmahone

    MHB Convergence of $\displaystyle\sum\frac{n^5}{2^n}$

    Does the following series converge? $\displaystyle\sum\frac{n^5}{2^n}$
  25. alexmahone

    MHB When does the series $\sum_2^\infty\frac{1}{n(\ln n)^p}$ diverge for $p<0$?

    Test for convergence: $\sum_2^\infty\frac{1}{n(\ln n)^p}$ when $p<0$. My working: Consider the function $f(x)=\frac{1}{x(\ln x)^p}=\frac{(\ln x)^q}{x}$ when $q=-p>0$. In order to use the integral test, I have to establish that $f(x)$ is decreasing for $x\ge 2$. How do I proceed?
  26. alexmahone

    MHB Does the series $\sum_1\sin\left(\frac{1}{n}\right)$ converge?

    Test for convergence: $\sum_1\sin\left(\frac{1}{n}\right)$ My attempt: $\sin\left(\frac{1}{n}\right)\sim\frac{1}{n}$ Since $\sum_1\frac{1}{n}$ diverges, $\sum_1\sin\left(\frac{1}{n}\right)$ also diverges by the asymptotic convergence test. Is that correct?
  27. alexmahone

    MHB Subsequence - absolute convergence

    Let $\{a_n\}$ be a sequence, and $\{a_{n_i}\}$ be any subsequence. Prove that if $\sum_{n=0}^\infty a_n$ is absolutely convergent, then $\sum_{i=0}^\infty a_{n_i}$ is absolutely convergent. My attempt: $\sum |\ a_n|$ is convergent. $b_n=\left\{ \begin{array}{rcl}|a_{n_i}|\ &\text{for}& \...
  28. W

    Convergence in Galilean space-time

    To talk about differentiable vector fields in Galilean space-time, one needs to define convergence. Galilean space-time is an affine space and its associated vector space is a real 4-dimensional vector space which has a 3-dimensional subspace isomorphic to Euclidean vector space. There is no...
  29. alexmahone

    MHB Is the Absolute Convergence Theorem Sufficient to Prove This?

    Prove: if $\sum a_n$ is absolutely convergent and $\{b_n\}$ is bounded, then $\sum a_nb_n$ is convergent. My working: $|b_n|\le B$ for some $B\ge 0$. $|a_n||b_n|<B|a_n|$ Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges. So, $\sum a_nb_n$ converges. (Absolute convergence...
  30. J

    Complex Power Series Radius of Convergence Proof

    Homework Statement If f(z) = \sum an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0. Homework Equations The Attempt at a Solution I know it is a power series and because R is positive I know it converges. And if...
  31. G

    Interesting convergence of sequence

    Homework Statement Let (a_n)_{n\in\mathbb{N}} be a real sequence such that a_0\in(0,1) and a_{n+1}=a_n-a_n^2 Does \lim_{n\rightarrow\infty}na_n exist? If yes, calculate it. Homework Equations The Attempt at a Solution I have a solution but I'd like to see other solutions..
  32. M

    Understanding L1 vs. Lp Weak Convergence

    Hello, I am struggling to understand why L1 is not weakly compact, while Lp, p>1, is. The example I have seen put forward is the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, the function being defined on (0,1). It is shown this u_n converging to the Dirac measure, and...
  33. M

    Convergence Induction for Positive Sequences: Proving Limit Behavior

    Homework Statement If (an) — > 0 and \bn - b\ < an, then show that (bn) — > b Homework Equations The Attempt at a Solution
  34. C

    Does this sequence converge to the proposed limit?

    Homework Statement Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. a) lim \frac{1}{6n^2+1}=0 b) lim \frac{3n+1}{2n+5}=\frac{3}{2} c) lim \frac{2}{\sqrt{n+3}} = 0 The Attempt at a Solution A sequence a_n...
  35. N

    Order of Convergence: Iteration Scheme

    Homework Statement I have attached the actual problem as a picture (hope that is ok). Xn+1 = 12/(1+Xn), r=3 For each of the following iteration schemes the sequence xn will converge to the given value of r provided the initial value x0 is sufficiently close to r. In each case determine the...
  36. K

    Pointwise Convergence For Induced Distributions

    Homework Statement If U \subseteq \mathbb R^n find a sequence of locally integrable functions f_n \in L^1_{\text{loc}}(U) which converge pointwise, but whose induced distributions \langle f_n, \cdot \rangle: C_c^\infty(U) \to \mathbb R, \qquad \langle f_n, \phi \rangle = \int_U f_n \phi...
  37. M

    Test for convergence of a series

    Homework Statement Homework Equations The Attempt at a Solution I have no ideas how to continue. I also tried the comparison test but I don't know where to start. Please guide me...
  38. Also sprach Zarathustra

    MHB Uniform Convergence of $Y(x)$ in $(0,1]$

    Hello!A little problem:With the given series,$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?Ok, I know that $Y(x)$ is u.c by M-test: $$\max{|x\ln{x}|}=\frac{1}{e}$$ And, $$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$ Is...
  39. F

    Convergence of $\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n (z\in C)

    \sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n z in C. This only converges for z>\frac{-1}{2}, correct? Thanks.
  40. S

    Proving Convergence of {S_n/n} for Bounded Sequence {S_n}

    Homework Statement If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent. We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze...
  41. C

    Complex Convergence with Usual Norm

    Homework Statement Determine whether the following sequence {xn} converges in ℂunder the usual norm. x_{n}=n(e^{\frac{2i\pi}{n}}-1) Homework Equations e^{i\pi}=cos(x)+isin(x) ε, \delta Definition of convergence The Attempt at a Solution I would like some verification that this...
  42. F

    Radius of Convergence for \sum_{n=2}^{\infty}z^n\log^2(n) in Complex Numbers

    \sum_{n=2}^{\infty}z^n\log^2(n), \ \text{where} \ z\in\mathbb{C} \sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2) By the ratio test, \lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right| \lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log...
  43. T

    I think this is a dominated convergence theorem question

    Hey All, I have the following integral expression: y = lim_{h\to0^{+}} \frac{1}{2\pi} \left\{\int^{\infty}_{-\infty} P(\omega)\left[e^{i\omega h} - 1 \right] \right\} \Bigg/ h And I am trying to understand when this expression will be zero. I was talking to a mathematician who said...
  44. estro

    Check Uniform Convergence of f_n(x): Solutions & Explanations

    Homework Statement f(x) is defined within [a,b]. f_n(x)=\frac{\big\lfloor nf(x) \big\rfloor}{n} Check if f_n(x) is uniform convergent. The Attempt at a Solution This one seems to be easy however since I didn't touch calculus for quite a time I'm not confident with my solution. |\frac...
  45. L

    Show Convergence of sequence (1 + c)(1 + c^2) (1 + c^n)

    Homework Statement Hey, so here is the problem: Suppose 0≤c<1 and let an = (1 + c)(1 + c2)...(1 + cn) for integer n≥1. Show that this sequence is convergent. Well I understand the basic concepts of proving convergence of sequences, but in class we've only ever done it with sequences where...
  46. C

    Convergence of {fn} wrt to C(X) metric

    Let C(X) be the set of continuous complex-valued bounded functions with domain X. Let {fn} be a sequence of functions on C. We say, that fn converges to f wrt to the metric of C(X) iff fn converges to f uniformly. By definition of the uniform convergence, for any ε>0 there exists integer N...
  47. A

    Trying to Prove Uniform Convergence: Analysis II

    Homework Statement I have a solution to the following problem. I feel it is somewhat questionable though If fn converges uniformly to f, i.e. fn\rightarrowf as n\rightarrow∞ and gn converges uniformly to g, i.e. gn\rightarrowf as n\rightarrow∞ , Prove that fngn...
  48. A

    Comparing Convergence: Exploring the p-Series Divergence for p<1

    Homework Statement \sum\frac{1}{n^{p}} converges for p>1 and diverges for p<1, p\geq0. The Attempt at a Solution (1) Diverges: I want to prove it diverges for 1-p and using the comparison test show it also diverges for p. \sum\frac{1}{n^{1-p}}=\sum\frac{1}{n^{1}n^{-p}}=\sum...
  49. C

    Show convergence using comparison test on sin(1/n)

    Homework Statement a) Test the following series for convergence using the comparison test : sin(1/n) Explain your conclusion. Homework Equations The Attempt at a Solution i must show f(x)<g(x) in order for it to converge other wise divergence. g(x) = 1/n sin(1/n) >...
  50. J

    Domain of convergence of Puiseux series

    May I ask if it's appropriate to ask in this sub-forum, are there any general methods for computing the domain of convergence for Puiseux series representations of algebraic varieties? I'm unable to find anything on the matter. For example, suppose I'm given the ideal in \mathbb{C}[z,w]...
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