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. 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...
  2. 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...
  3. 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}\\...
  4. 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.
  5. 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|...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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 >...
  14. 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...
  15. 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...
  16. 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...
  17. 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 $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$...
  18. 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 $a<-1$ ...
  19. 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 =...
  20. 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...
  21. 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} <...
  22. 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...
  23. 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) +...
  24. 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...
  25. 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...
  26. 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 ##...
  27. 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?
  28. 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...
  29. 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...
  30. 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?
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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##...
  37. ertagon2

    MHB Sequences and their limits, convergence, supremum etc.

    Could someone check if my answers are right and help me with question 5?
  38. 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)...
  39. 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...
  40. F

    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...
  41. 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...
  42. 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...
  43. 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))...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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|...
  50. 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...
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