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

    Uniform convergence of a sequence of functions

    Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
  2. F

    Showing Uniform Convergence of Cauchy Sequence of Functions

    Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...
  3. N

    Interval of uniform convergence of a series

    Homework Statement The series is uniformly convergent on what interval? Homework EquationsThe Attempt at a Solution [/B] Using the quotient test (or radio test), ##|\frac{a_{n+1}}{a_{n}}| \rightarrow |x^2*\sin(\frac{\pi \cdot x}{2})|, n \rightarrow \infty##. However from here I'm stuck...
  4. A

    Show that the integral converges

    Homework Statement (FYI It's from an Real Analysis class.) Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent. Homework Equations I know that for an integral to be convergent, it means that : $$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.I can also use the...
  5. evinda

    MHB Are my ideas as for the convergence right?

    Hello! (Wave) I am looking at the following exercise: Let $(a_n)$ be a sequence of real numbers such that $a_{2n} \to a$ and $a_{2n+1} \to a$ for some real number $a$. Show that $a_n \to a$. We are given the sequence $x_n=0$ if $n$ is even, $x_n=1$ if $n$ is odd. Check as for the...
  6. evinda

    MHB Convergence and existence of constants

    Hello! (Wave) Let $m$ be a natural number. I want to check the sequence $\left( \binom{n}{m} n^{-m}\right)$ as for the convergence and I want to show that there exist constants $C_1>0, C_2>0$ (independent of $n$) and a positive integer $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$...
  7. evinda

    MHB Does the sequence $(a^n b^{n^2})$ converge for all values of $a$ and $b$?

    Hello! (Wave) I want to check as for the convergence the sequence $(a^n b^{n^2})$ for all the possible values that $a,b$ take. I have thought the following: We have that $\lim_{n \to +\infty} a^n=+\infty$ if $a>1$, $\lim_{n \to +\infty} a^n=0$ if $-1<a<1$, right? What happens for $a<-1$ ...
  8. Mr Davis 97

    Find which initial conditions lead to convergence

    Homework Statement Let ##b_1\in \mathbb{R}## be given and ##n=1,2,\dots## let $$b_{n+1} := \frac{1+b_n^2}{2}.$$ Define the set $$B := \{b_1\in\mathbb{R} \mid \lim_{n\to\infty}b_n \text{ converges}\}$$ Identify the set ##B##. Homework EquationsThe Attempt at a Solution I claim that ##B =...
  9. Mr Davis 97

    Convergence of oscillatory/geometric series

    Homework Statement Determine for which ##r\not = 0## the series ##\displaystyle {\sum_{n=1}^\infty(2+\sin(\frac{n\pi}{3})) r^n}## converges. Homework EquationsThe Attempt at a Solution We have to split this up by cases based on ##r##. 1) Suppose that ##0<|r|<1##. Then...
  10. Mr Davis 97

    Convergence of the Sequence √n(√(n+1)-√n) to 1/2

    Homework Statement Show that ##\displaystyle \lim_{n\to \infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n}) = \frac{1}{2}## Homework EquationsThe Attempt at a Solution We see that ##\displaystyle \sqrt{n}(\sqrt{n+1}-\sqrt{n}) - \frac{1}{2} = \frac{\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} - \frac{1}{2} <...
  11. Mr Davis 97

    Convergence of a recursive sequence

    Homework Statement With ##a_1\in\mathbb{N}## given, define ##\displaystyle {\{a_n\}_{n=1}^\infty}\subset\mathbb{R}## by ##\displaystyle {a_{n+1}:=\frac{1+a_n^2}{2}}##, for all ##n\in\mathbb{N}##.Homework EquationsThe Attempt at a Solution We claim that with ##a_1 \in \mathbb{N}##, the sequence...
  12. Felipe Lincoln

    Convergence of series log(1-1/n^2)

    Homework Statement Find the sum of ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) ## Homework Equations No one. The Attempt at a Solution At first I though it as a telescopic serie: ##\sum\limits_{n=2}^{\infty}\ln\left(1-\dfrac{1}{n^2}\right) =\ln\left(\dfrac{3}{4}\right) +...
  13. Felipe Lincoln

    Convergence of the series nx^n

    Homework Statement By finding a closed formula for the nth partial sum ##s_n##, show that the series ## s=\sum\limits_{n=1}^{\infty}nx^n## converges to ##\dfrac{x}{(1-x)^2}## when ##|x|<1## and diverges otherwise. Homework Equations Maybe ##s=\sum\limits_{n=0}^{\infty}x^n=\dfrac{1}{1-x}## when...
  14. FallenApple

    No convergence in scientific theories to a grand truth

    I thought about something interesting. Essentially any scientific theory is just a subset of the powerset of all possible human thoughts. Good theories are just stories within that powerset that happen to have predictive power and hence are useful to us. But the powerset of all possible human...
  15. chwala

    Does the Series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## Converge?

    Homework Statement Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or notHomework EquationsThe Attempt at a Solution looking at ## 1/sin (n) ## by comparison, ##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1## for ##n≥1 ## implying that ##{sin (n)}≤n ##...
  16. chwala

    Finding the convergence of a binomial expansion

    Homework Statement Expand ##(1+3x-4x^2)^{0.5}/(1-2x)^2## find its convergence valueHomework EquationsThe Attempt at a Solution on expansion ##(1+3/2x-3.125x^2+4.6875x^3+...)(1+4x+12x^2+32x^3+...)## ##1+5.5x+14.875x^2+42.1875x^3+... ## how do i prove for convergence here?
  17. L

    A Convergence of a subsequence of a sum of iid r.v.s

    ##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
  18. Peter Alexander

    Uniform convergence of a parameter-dependent integral

    Hello everyone! I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...
  19. mjtsquared

    I Region of convergence of a Laplace transform

    If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?
  20. J

    I Is the Arctan Convergence Rate Claim Valid for Positive Values of a, b, and x?

    What do you think about the claim that \frac{x}{\frac{1}{a} + \frac{x}{b}} \;<\; \frac{2b}{\pi}\arctan\Big(\frac{\pi a}{2b}x\Big),\quad\quad\forall\; a,b,x>0 First I thought that if this is incorrect, then it would be a simple thing to find a numerical point that proves it, and also that if...
  21. isukatphysics69

    Is the Interval of Convergence for (x-2)^n / n^(3n) from -1 to 5?

    Homework Statement interval of convergence for n=1 to inf (x-2)n / n3n Homework EquationsThe Attempt at a Solution i used the ratio test and solved for x and got that the interval of convergence is from -1 to 5. now i have to test the endpoints to determine which ones will make the series...
  22. isukatphysics69

    Radius of convergence of the power series (2x)^n/n

    Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...
  23. W

    I Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##

    Hi Physics Forums, I have a problem that I am unable to resolve. The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to...
  24. isukatphysics69

    Determine convergence of 2*abs(an) from 1 to inf

    Homework Statement Homework EquationsThe Attempt at a Solution confused here, so my book seems to be saying that 2*abs(an) converges which i thought was bogus so i went over to symbolab and symbolab is saying it diverges which i agree with. Why is my book saying this? Am i misenterpereting...
  25. Poetria

    Addition of power series and radius of convergence

    Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...
  26. ertagon2

    MHB Sequences and their limits, convergence, supremum etc.

    Could someone check if my answers are right and help me with question 5?
  27. binbagsss

    Triangle Inequality: use to prove convergence

    Homework Statement Attached I understand the first bound but not the second. I am fine with the rest of the derivation that follows after these bounds, Homework Equations I have this as the triangle inequality with a '+' sign enabling me to bound from above: ##|x+y| \leq |x|+|y| ## (1)...
  28. Rectifier

    Pointwise vs. uniform convergence

    The problem I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##. Relevant equations ##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1) ##f_n## converges uniformly if ## \lim_{n...
  29. fresh_42

    I Weak Convergence of a Certain Sequence of Functions

    Given a function in ##f \in L_2(\mathbb{R})-\{0\}## which is non-negative almost everywhere. Then ##w-lim_{n \to \infty} f_n = 0## with ##f_n(x):=f(x-n)##. Why? ##f\in L_2(\mathbb{R})## means ##f## is Lebesgue square integrable, i.e. ##\int_\mathbb{R} |f(x)|^2 \,dx< \infty ##. Weak convergence...
  30. D

    Calculus II: Convergence of Series with Positive Terms

    Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...
  31. F

    Convergence of a continuous function related to a monotonic sequence

    Homework Statement Let ##f## be a real-valued function with ##\operatorname{dom}(f) \subset \mathbb{R}##. Prove ##f## is continuous at ##x_0## if and only if, for every monotonic sequence ##(x_n)## in ##\operatorname{dom}(f)## converging to ##x_0##, we have ##\lim f(x_n) = f(x_0)##. Hint: Don't...
  32. E

    Can you help me determine the convergence of these series?

    Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...
  33. Cathr

    Improper integral convergence from 0 to 1

    Homework Statement I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent. Homework Equations I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1. The Attempt at a Solution The function can be approximated to ln(x) when it approaches...
  34. M

    MHB Convergence of iteration method - Relation between norm and eigenvalue

    Hey! :o Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ? I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or...
  35. Mr Davis 97

    I Convergence of a recursively defined sequence

    I have the following sequence: ##s_1 = 5## and ##\displaystyle s_n = \frac{s_{n-1}^2+5}{2 s_{n-1}}##. To prove that the sequence converges, my textbook proves that the following is true all ##n##: ##\sqrt{5} < s_{n+1} < s_n \le 5##. I know to prove that this recursively defined sequence...
  36. D

    Convergence of a series with n-th term defined piecewise

    Homework Statement Test the series for convergence or divergence ##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...## Homework Equations rn=abs(an+1/an) The Attempt at a Solution With some effort I was able to figure out the 'n' th tern of the series an = \begin{cases} 2^{-(0.5n+1.5)} & \text{if } n...
  37. Mr Davis 97

    Convergence of Sequence Proof: Is This Correct?

    Homework Statement Prove rigorously that ##\displaystyle \lim \frac{n}{n^2 + 1} = 0##. Homework Equations A sequence ##(s_n)## converges to ##s## if ##\forall \epsilon > 0 \exists N \in \mathbb{N} \forall n \in \mathbb{N} (n> N \implies |s_n - s| < \epsilon)## The Attempt at a Solution Let...
  38. C

    MHB Show that a sequence is bounded, monotone, using The Convergence Theorem

    Dear Every one, In my book, Basic Analysis by Jiri Lebel, the exercise states "show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit" My Work: The Proof: Bound The sequence is bounded by 0. $\left|{(n+1)/n}\right|...
  39. Math Amateur

    MHB Coordinate-Wise Convergence in R^n .... TB&B Chapter 11, Section 11.4 ....

    I am reading the book "Elementary Real Analysis" (Second Edition, 2008) Volume II by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner ... and am currently focused on Chapter 11, The Euclidean Spaces \mathbb{R}^n ... ... I need with the proof of Theorem 11.15 on coordinate-wise...
  40. C

    Finite element solving of Laplace's equation doesn't converge

    Homework Statement I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
  41. S

    Interval of convergence

    Homework Statement Find the interval of convergence of: ##\sum\frac{n^n}{n!}z^n## Homework EquationsThe Attempt at a Solution I obtained that the radius of convergence is ##1/e## but I am not sure what to do at the end points. For ##z=1/e## I would have ##\sum{n^n}{n!e^n}##. Mod edit: I think...
  42. S

    Convergence of a double summation using diagonals

    Homework Statement Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##. Homework Equations I've included some relevant information below: The Attempt at a Solution So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
  43. E

    MHB Prove limit with convergence tests

    I need to prove that the limit of the sequence is as shown(0): 1.limn→∞ n*q^n=0,|q|<1 2.limn→∞ 2*n/n! but I need to do this using the convergence tests. With the second sequence I tried the "ratio test", and I got the result limn→∞ 2/n+1 which means that L in the ratio test is 0 and so it...
  44. Delta2

    I Rational sequence converging to irrational

    In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that "for any irrational number there exists a sequence of rational numbers that converges to it", and it doesn't have a proof for it, just saying that it is a...
  45. S

    Ratio Test and Radius of Convergence for ∑ ((n-2)2)/n2, n=1: Homework Solution

    Homework Statement ∞ ∑ = ((n-2)2)/n2 n=1 Homework Equations The ratio test/interval of convergence The Attempt at a Solution **NOTE this is a bonus homework and I've only had internet tutorials regarding the ratio test/interval of convergence so bear with me) lim ((n-1)n+1)/(n+1)n+1 *...
  46. T

    MHB Power Series Convergence Assistance

    The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.
  47. R

    Proving the convergence of series

    Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
  48. L

    A Convergence order of central finite difference scheme

    For example, when we solve simple 1D Poisson equation by finite difference method, why three point central difference scheme on uniform grid (attached image) is second order method for solution convergence? I understand why approximation of first derivative is second order (and that second...
  49. T

    Refraction Convergence and Amplitude change- Ocean waves

    There are many explanations on the internet, of refraction and convergence of ocean waves entering shallow water around a headland However they all go no deeper than this statement "Where the water is shallow the wave rays converge wave energy is greater where the wave rays spread out the...
  50. Euler2718

    Proof of sequence convergence via the "ε-N" definition

    Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...
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