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. C

    Alternating Series Convergence Test

    According to my calculus book two parts to testing an alternating series for convergence. Let s = Ʃ(-1)n bn. The first is that bn + 1 < bn. The second is that the limn\rightarrow∞ bn = 0. However, isn't the first condition unnecessary since bn must be decreasing if the limit is zero. I...
  2. B

    Convergence of this sequence .

    Homework Statement find the limit n\rightarrow∞ of 10n/ n! Homework Equations L hospital rule The Attempt at a Solution took log and separated the num and denom as: n ln10-ln(n!) n ln10-n ln(n)+n 1/n ( ln10 - ln(n)+1) now i...
  3. D

    Ratio test for math convergence

    Homework Statement show ## \sum \frac{x^{2}}{(1+x^{2})^{n}} ## converges uniformly on R Homework Equations The Attempt at a Solution I know by ratio test it is absolutely convergent for all x in R. I am guessing you use m-test. However I do not really understand how...
  4. C

    Why Does My LED Array Convergence Fail in Falstad's Circuit Simulator?

    im using falstad and my knowledge on circuits is poor or extremely basic.. $ 1 5.0E-6 34.14951009862697 50 5.0 50 162 512 288 816 288 1 2.1024259 1.0 0.0 0.0 162 544 288 896 288 1 2.1024259 1.0 0.0 0.0 162 576 288 960 288 1 2.1024259 1.0 0.0 0.0 w 896 288 816 288 0 g 960 288 960 240 0...
  5. A

    Does Using Maximum Coefficients Determine the Smallest Radius of Convergence?

    Homework Statement Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B) Homework Equations My idea is to use that the series...
  6. A

    Determining Convergence of Series Using Comparison and Ratio Tests

    Homework Statement Does the series \Big( \sum_{n=1}^\infty\frac{1}{(3^n)*(sqrtn)} \Big) Converge or Diverge? By what test?Homework Equations 1/n^p If p<1 or p=1, the series diverges. If p>1, the series converges. If bn > an and bn converges, then an also converges. The Attempt at a...
  7. O

    Convergence Test (Comparison) Questions

    Hello everyone, I need some help on doing convergence tests (comparisons I believe) on some Ʃ sums. I have three, they are: 1. Ʃ [ln(n)/n^2] from n=1 to ∞. I tried the integral test but was solved to be invalid (that is, cannot divide by infinity). Therefore I believe it to be a...
  8. S

    MHB Glad to hear that the answer is correct! You are welcome, happy to help.

    |f(x0)f''(x0)|<|f'(x0)|^2 where I is the interval containing the approximate root x0, is the convergence criterion of ... (a) Newton - Raphson method (b) Iteration method (c) Secant method (d) False position method According to me its (a), but I confused because this formula is not directly...
  9. alyafey22

    MHB Discussions on the convergence of integrals and series

    This thread will be dedicated to discuss the convergence of various definite integrals and infinite series , if you have any question to post , please don't hesitate , I hope someone make the thread sticky. 1- \int^{\infty}_0 \left(\frac{e^{-x}}{x} \,-\,\frac{1}{x(x+1)^2}\right)\,dx\,=1-\gamma...
  10. Fernando Revilla

    MHB Emil's question at Yahoo Answers (Radius of convergence)

    Here is the question: Here is a link to the question: How to find Radius of Convergence for Sum of ((x-3)^n)/(n3^n) from n =1 to inf? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  11. alyafey22

    MHB Does the integral \int^{\infty}_{0} \frac{\log(t) \sin(t) }{t} \, dt converge?

    \int^{\infty}_{0} \frac{\log(t) \sin(t) }{t} \, dt Can we say the following : \int^{\infty}_{0} \frac{\log(t) \sin(t) }{t} \, dt=\int^{\epsilon}_{0} \frac{\log(t) \sin(t) }{t} \, dt+\int^{\infty}_{\epsilon} \frac{\log(t) \sin(t) }{t} \, dt 1-\int^{\epsilon}_{0} \frac{\log(t) \sin(t) }{t} \...
  12. P

    Find the interval of convergence of (x^n)/(n +1)

    Homework Statement My main issue here is that if I use the ratio test I end up with lim (x(n!+1))/((n+1)!+1) n-> ∞ I don't know how to progress here. I believe that the limit will equal 0 and so it's interval of convergence is from -∞<x<∞ with a Radius of convergence of ∞. Is it safe for me...
  13. B

    MHB Convergence and Divergence - Calculus 2

    Alright, I have my final for Calc 2 on Monday. I am only stuck on two sections of problems because I am terrible at convergence and divergence/alternating series. I have questions for each one really. They are as follows: 4) a. Can I simply factor out the alternating series and apply a...
  14. P

    Radius and Interval of convergence

    Homework Statement Ʃ(((-1)^n)(x^n))/(n+1) from 0 to ∞Homework Equations The Attempt at a Solution I took applied the ratio test and got that lim|(x^(n+1))/(n+2) * (n+1)/(x^n)| =|x| so that means for it to converge |x|<1 Radius of convergence is 1 My interval is (-1<x<1) Now I check the...
  15. polygamma

    MHB Convergence of integral for $0 \le a <1$

    How do you show that $\displaystyle \int_{0}^{\infty} \Big( \frac{x^{a-1}}{\sinh x} - x^{a-2} \Big) \ dx $ converges for $0 \le a <1$?
  16. F

    Sequence Convergence & Fixed Point Theorem

    Homework Statement Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1 a) Determine the values of p0 \in [1,2] for which the sequence {pn} from 0 to infinity converges. b) For the cases where {pn} converges (if any), what is the rate of convergence...
  17. M

    Prime Number Series Convergence to p-adic Convergence | Homework Help"

    Homework Statement Let p be a prime number. Which of the following series converge p-adically? Justify your answers: (all sums are from n = 0 to infinity) (i) Ʃp^n (ii) Ʃp^-n (iii) Ʃn! (iv) Ʃ (2n)! / n! (v) Ʃ (2n)! / (n!)^2 Homework Equations The definition given for p-adic...
  18. A

    Radius of convergence of power series

    Homework Statement The coefficients of the power series \sum_{n=0}^{∞}a_{n}(x-2)^{n} satisfy a_{0} = 5 and a_{n} = (\frac{2n+1}{3n-1})a_{n-1} for all n ≥ 1 . The radius of convergence of the series is: (a) 0 (b) \frac{2}{3} (c) \frac{3}{2} (d) 2 (e) infinite Homework EquationsThe Attempt at...
  19. D

    Proving Convergence of Series: (a_n) and (a_{2n-1} + a_{2n})

    Homework Statement Let (a_n) be a sequence. (i) Prove that if \sum\limits_{n = 1}^\infty {{a_n}} converges, then \sum\limits_{n = 1}^\infty {\left( {{a_{2n - 1}} + {a_{2n}}} \right)} also converges. (ii) Prove that if \sum\limits_{n = 1}^\infty {\left( {{a_{2n - 1}} + {a_{2n}}} \right)}...
  20. Fernando Revilla

    MHB Joanne 's question at Yahoo Answers (Interval of convergence)

    Here is the question: Here is a link to the question: Determine the interval of convergence? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  21. H

    What is the Proof for Locally Uniformly Convergence of Series in the Unit Disc?

    Homework Statement Let f(z) be holomorphic in the unit disc B(0,1),such thaf f(0)=0.Prove that the series Ʃf(z^n) is locally uniformly convergence in B(0,1).Homework Equations locally uniformly convergence:if it is uniformly convergence in a neibourhood of each point of B(0,1).The Attempt at a...
  22. P

    Finding interval of convergence

    1. Determine a power series, centered at zero for the function ∫f(x)dx. Identify the interval of convergence. f(x) = ln(x+1) = ∫\frac{1}{x+1} 2. 3. i found the power series, which is : Ʃ ((-1)^(n))(x^(n+1)) / (n+1) Im okay with that, but i need help on finding the interval of...
  23. R

    A proof for Gauss' test for convergence

    Homework Statement if \frac{a_n}{a_{n+1}}=1+\frac{p}{n}+β_n and β converges absolutely, then at the infinity the sequence a is of the same order as \frac{c}{n^p}. Homework Equations Basic convergence test(Cauchy and d'Alembert's tests) The Attempt at a Solution I find this question...
  24. T

    MHB Infinite series convergence III

    Test these for convergence. 5. infinity E...((n!)^2((2n)!)^2)/((n^2 + 2n)!(n + 1)!) n = 0 6. infinity E...(1 - e ^ -((n^2 + 3n))/n)/(n^2) n = 3 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  25. T

    MHB Infinite series convergence II

    Test these for convergence. 3. infinity E...((-1)^n)*(n^3 + 3n)/((n^2) + 7n) n = 2 4. infinity E...ln(n^3)/n^2 n = 2 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  26. T

    MHB Do These Infinite Series Converge?

    Test these for convergence. 1. infinity E...n!/(n! + 3^n) n = 0 2. infinity E...(n - (1/n))^-n n = 1 Btw, E means sum. Which tests should I use to solve these?
  27. I

    Proof of convergence theory in optimization

    Homework Statement The question is: Suppose that lim x_k=x_*, where x_* is a local minimizer of the nonlinear function f. Assume that \triangledown^2 f(x_*) is symmetric positive definite. Prove that the sequence \left \{ f(x_k)-f(x_*) \right \} converges linearly if and only if \left...
  28. I

    MHB Proof of convergence theory in optimization

    The question is:Suppose that lim $x_k=x_*$, where $x_*$ is a local minimizer of the nonlinear function $f$. Assume that $\triangledown^2 f(x_*)$ is symmetric positive definite. Prove that the sequence $\left \{ f(x_k)-f(x_*) \right \}$ converges linearly if and only if $\left \{ ||x_k-x_*||...
  29. Fernando Revilla

    MHB Theodore K's question at Yahoo Answers (Radius of convergence)

    Here is the question: Here is a link to the question: Calculus Power Series/Radius of Convergence/Interval of Convergence Question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  30. K

    Investigating the Convergence of Series: Sn = 5-1/n

    Homework Statement The nth partial sum of the series ∞ Ʃ an n=1 is given Sn = 5-1/n Determine weather the series is convergent or divergent The Attempt at a Solution Looked in my book on how to do this one. couldn't find anything on it. What i was thinking...
  31. C

    Types of Convergence of the DTFT & Relation to Summability of x[n]

    Given a discrete time signal x[n] that has a DTFT (which exists in the mean square convergence or in the uniform convergence sense), how can we tell if the signal x[n] converges absolutely? I know the following: x[n] is absolutely summable <=> X(e^{j \omega}) converges uniformly (i.e...
  32. C

    Absolute convergence proving that limit =1

    Homework Statement limit as k--> infinity ABS VALUE((cos(pi*k+pi))/(cos(k*pi))) Homework Equations The Attempt at a Solution Can someone prove to me why this limit is equal to 1? I have tried several other sources and I have not had any luck.
  33. A

    Constructing a Sequence for Pointwise Convergence and Unboundedness

    Give an example of a sequence \{ f_n\} of continuous functions defined on [0,1] such that \{ f_n\} converges pointwise to the zero function on [0,1], but the sequence \{ \int^{1}_{0} f_n\} is unbounded. I'm pretty lost on this one.
  34. twoski

    What is the Interval of Convergence for (x-10)^n/10^n?

    Homework Statement Find the interval of convergence for (x-10)^{n}/10^{n} The Attempt at a Solution If i use the ratio test on this, i end up with (x-10)/10, which doesn't make sense to me since there is no "n" in this result. Is there another method i need to be using?
  35. A

    Problem in convergence in Quantum espresso

    I have constructed mono layer slab of rutile TiO2 (110) with 11 Angstroms vacuum between two layers. I used the Quantum wise software for the construction of the slab. Then I have exported the Quantum espresso input from Quantum wise software. Now the problem is the SCF calculations are not...
  36. W

    Does the given series converge absolutely or conditionally?

    Homework Statement Determine either absolute convergence, conditional convergence or divergence for the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}The Attempt at a Solution It converges conditionally i know, but i can't figure out how. 1. I applied the...
  37. W

    Absolute and Conditional Convergence Tests for Series | Homework Equations

    Homework Statement Absolute, Conditional, - convergence, or Divergence.Homework Equations \displaystyle \sum^{∞}_{n=1} (-1)^n e^{-n} The Attempt at a Solution 1. Alternating Series Test 2. Ratio Test for ABsolute Convergence 1. \displaystyle (-1)^n (1/e)^n an > 0 for n=1,2,3,4 - YES...
  38. W

    Find the Interval of Convergence

    Homework Statement Find the interval of convergence.Homework Equations \displaystyle \sum^{∞}_{n=0} \frac{(x-5)^n}{n^4 * 2^n}The Attempt at a Solution I used the ratio test as follows: \displaystyle \frac{(x-5)^{n+1}}{(n+1)^4 * 2^{n+1}} * \frac{n^4 2^n}{(x-5)^n} taking the limit: (x-5) lim...
  39. X

    Radius of Convergence Power Series

    Homework Statement Determine the radius of convergence and the interval of convergence og the folling power series: n=0 to infinity Ʃ=\frac{(2x-3)^{n}}{ln(2n+3)} Homework Equations Ratio Test The Attempt at a Solution Well I started with the ratio test but I have no clue where...
  40. C

    MHB Ratio Test Questions/ Series Convergence

    I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
  41. STEMucator

    Deceptive uniform convergence question

    Homework Statement http://gyazo.com/55eaace8994d246974ef750ebeb36069 Homework Equations Theorem III : http://gyazo.com/af2dfeb33d3382430d39f275268c15b1 The Attempt at a Solution At first this question had me jumping to a wrong conclusion. Upon closer inspection I see the...
  42. C

    MHB Determining convergence of series

    I have a question on which test to use for series n=1 to infinity for (-1)^n / (n^3)-ln(n) in order to determine convergence/divergence. I am pretty sure I determined it converges through the Alternating Series Test(correct me if I'm wrong) but I am not sure whether it is conditional or...
  43. T

    Problem with function of sequence simple convergence

    Hello. I have a problem. I don't know how to finish my proof. Maybe someone of you will be able to help me. So task is: Suppose that fn and gn are function of the sequence. fn simply convergence(? don't know if english term is same) to f, and gn simply convergence to g. Need to prove that...
  44. STEMucator

    Proving Uniform Convergence of ##s_n(x)## to ##s(x)## on ##[b, ∞)##

    Homework Statement Suppose that ##s_n(x)## converges uniformly to ##s(x)## on ##[b, ∞)##. If ##lim_{x→∞} s_n(x) = a_n## for each n and ##lim_{n→∞} a_n = a## prove that : ##lim_{x→∞} s(x) = a## Homework Equations ##\space ε/N## The Attempt at a Solution I see a quick way to do this one...
  45. Astrum

    What is the Intuitive Explanation for the Definition of Convergence?

    I'm a bit confused about how my book defines convergence. Definition: A sequence {an} convergences to l if for every ε > 0 there is a natural number N such that, for all natural numbers n, if n > N, then l a,-l l < ε note, l a,-l l = the absolute value Maybe someone could give me an...
  46. B

    Convergence of Sequence ##(p_n)## to ##p##

    If ##p## is a limit point of ##E## then ##\exists \ (p_n) \ s.t. (p_n) \rightarrow p## For the sequence construction, can I just define ##(p_n)## as such: ##For \ q \in E, \ \ define \ (p_n) := \left\{\Large{\frac{d(p,q)}{n}} \right\}_{n=1}^\infty##
  47. P

    Challenging integrals/series convergence problems

    I'm not sure if this should go in the homework section, since I'm essentially asking for textbook style problems. Anyways, if you know of any good integrals/series convergence problems off the top of your head, could you give some to me? I guess if I can't figure any out I'll post them as...
  48. D

    Construct extrapolation table with optimal convergence

    Construct extrapolation table with optimal rates of convergence Homework Statement Let S be a cubic spline interpolant that approximates a function f on the given nodes x_{0},x_{1},...,x_{n} with the boundary conditions: S''(x_{0})=0 and S'(x_{n})=f'(x_{n}). Use S to estimate f(0.1234567)...
  49. W

    Seq Convergence: Does a_n = (-1)^(n+1)/(2n-1) Converge or Diverge?

    Homework Statement Converge or diverge?Homework Equations \displaystyle a_{n} = \frac{(-1)^{n+1}}{2n-1}The Attempt at a Solution all i know is to perhaps take ln of numerator and denominator to get the exponent down below?
  50. STEMucator

    Quick question about convergence

    Homework Statement Let ##s_n(x) = \frac{1}{n} e^{-(nx)^2}##. Show there is a function ##s(x)## such that ##s_n(x) → s(x)## uniformly on ##ℝ## and that ##s_n'(x) → s'(x)## for every x, but that the convergence of the derivatives is not uniform in any interval which contains the origin...
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