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. K

    Random Variables: Convergence in Probability?

    Definition: Let X1,X2,... be a sequence of random variables defined on a sample space S. We say that Xn converges to a random variable X in probability if for each ε>0, P(|Xn-X|≥ε)->0 as n->∞. ==================================== Now I don't really understand the meaning of |Xn-X| used in...
  2. T

    Proving Absolute Convergence of Gamma and Beta Integrals in Complex Analysis

    Homework Statement Let z,p,q \in \mathbb{C} be complex parameters. Determine that the Gamma and Beta integrals: \displaystyle \Gamma (z) = \int_0^{\infty} t^{z-1} e^{-t}\;dt \displaystyle B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt converge absolutely for \text{Re}(z)>0 and p,q>0...
  3. Y

    Convergence of a sequence of points on a manifold

    I have a question regarding the following definition of convergence on manifold: Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i \phi_i(x_k)_{k>N}...
  4. I

    Proving Absolute Value Convergence of Sequence to A

    Homework Statement If the absolute value of a sequence, an converges to absolute value of A, does sequence, an necessarily converge to A? Homework Equations convergence: a sequence { an}n=1-->infinity, converges to A є R (A is called the limit of the sequence) iff for all є > 0, there...
  5. S

    Convergence of Sum 1/n(n+1) * (sin(x))^n

    Homework Statement Sum 1/n(n+1) * (sin(x))^n . Show this converges for all x in the reals. Find with proof an interval on which it determines a differentiable function of x together with an expression of its derivative in terms of standard functions. Homework Equations The...
  6. S

    Convergence of Infinite Sums of Trigonometric Functions: Finding the Range of x

    Homework Statement Find what range of values of x the infinite sum of sin2n(x) and infinite sum 2nsin2n-1(x) converge and find an expression for their sums, carefully justifying your answers. The Attempt at a Solution I used cauchys root testand basically got that the first sum...
  7. R

    Convergence in distribution

    Hi, Here is my question: Given that X_n\xrightarrow{\mathcal{D}}Z as n\rightarrow\infty where Z\sim N(0,1). Can we conclude directly that \lim_{n\rightarrow\infty}P(|X_n|\leq u)=P(|Z|\leq u) where u\in (0,1)? Is this completely trivial or requires some proof? Also what is the differences...
  8. M

    Proving convergence of factorial w/o Ratio Test

    Homework Statement Determine whether 1/n! diverges or converges. So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it. Homework Equations N/a The Attempt at a Solution I...
  9. S

    Convergence of a sum for which x?

    Homework Statement Consider the infinite series (1/n) * (xn) where x is a real noumber. Find all numbers x such that i) the series converges, ii) series converges absolutely iii) diverges to + infinity iiii) does not converge. 2. The attempt at a solution For this, i know it...
  10. I

    Convergence of densities in Lindeberg's CLT

    Hi there, The central limit theorem asserts that the normalized sum of a sequence of i.i.d. random variables X_1, X_2,..., with finite variance converges in distribution to a normal distribution. Moreover, there is a result by Ranga Rao which guarantees that if X_i has a pdf, then the...
  11. T

    Power Series: Interval Of Convergence

    Homework Statement I am not really good with Series so I having a hard time with these problems. http://img835.imageshack.us/img835/858/img1257d.jpg Homework Equations The Attempt at a Solution The part I am stuck is where I highlighted. The first question: The whole thing is squared so I...
  12. Q

    Finite Element Convergence Analysis

    Homework Statement I have a designed a Finite Element code for a Poisson problem using Bilinear Element. - \DeltaU= 2\pi^{2}Sin(\pi X)Sin(\pi Y) U = 0 The exact solution is given by : UEX = Sin(\pi X)Sin(\pi Y). 2. The attempt at a solution On convergence analysis:Theoretically, if...
  13. S

    Convergence of implicit Euler method

    Homework Statement The implicit Euler method is yn = yn-1 + hf(xn,yn). Find the local truncation error and hence show that the method is convergent. Homework Equations The Attempt at a Solution I found the error to be ln = (-h2/2)y''(xn-1) + O(h3). For convergence I am up to...
  14. K

    What is the operation between row 2 and row 3 in the convergence of this series?

    Homework Statement I need to examine convergence of series with a term Un given below. The solution is given, but I can't understand what happens between row 2 and row 3. What kind of operation is that, does it have something in common with Taylor series expansion...
  15. A

    Absolute Convergence, Conditional Convergence or divergence

    Absolute Convergence, Conditional Convergence or divergence... Homework Statement \sum_{n=1}^{\infty} \frac {(-2)^{n}}{n^{n}} Homework Equations \lim_{n \rightarrow \infty} | \frac {a_{n+1}}{a_n}| < 1 \;\; absolute\; convergence \lim_{n \rightarrow \infty} | \frac...
  16. jfy4

    Absolute convergence, boundedness, and multiplication

    Homework Statement If the series \sum_{n=1}^{\infty}x_n converges absolutely, and the sequence (y_n)_n is bounded, then the series \sum_{n=1}^{\infty}x_ny_n converges.Homework Equations Definitions and theorems relating to series and convergence.The Attempt at a Solution If the sequence y_n is...
  17. J

    Interval of Convergence and radicals

    Homework Statement Find the interval of convergence: \sum _{n=1}^{\infty } \frac{(-1)^n (x+2)^n}{3^n\sqrt{n}} Homework Equations The Attempt at a Solution \lim_{n\to \infty } |\frac{(x+2)^{n+1}}{3^{n+1}\sqrt{n+1}}*\frac{3^n\sqrt{n}}{(x+2)^n}| = \lim_{n\to \infty }...
  18. dextercioby

    Weak versus strong convergence

    So I've seen the distinction one makes in case of infinite-dimensional Hilbert spaces. Weak convergence versus strong convergence of sequences. I cannot think of an example of sequence of vectors in L^2(R) which converges with respect to the scalar product, but not with respect to the norm...
  19. G

    Power series and the interval of convergence

    Homework Statement I need help finding the interval of convergence for f(x) = 3/(1-x^4). I think that the summation would be \Sigma 3 (x^4n) from n=0 to infinity, but I'm not sure how to get the interval of convergence. Homework Equations f(x) = 3/(1-x^4) The Attempt at a Solution...
  20. D

    Showing the uniform convergence of a gaussian function-like series

    Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...
  21. M

    Convergence of Sequence to e and around e

    I was thinking of how ( 1 + (1/n) ) ^ n converges to e and I am aware of how if it is raised to some an, then it converges to e^a. If i recall if the form ( 1 + (a/n) ) ^ n converges to ae? I was hoping someone could tell me how to deal with ( 1 + (1/n^2) ) ^ n? Thanks!
  22. M

    Determine convergence for a series

    Homework Statement Sum ln n/(ln(ln n)) n=3..infinity? Im pretty sure it diverges and I am pretty sure to use the limit test but i just don't know what to compare this sum to. Would 1/n be ok. Do i have to justify why they are similar? ANy help would be nice thanks. Homework Equations...
  23. T

    Checking Convergence of PDF Integral

    Dear all, I have a PDF in independent variables q, \mu and h, all depending on x and y. I wish to check whether or not this PDF converges, which means checking that the normalisation constant converges in the limits -\infty and +\infty of the above mentioned variables q, \mu and h. The integral...
  24. L

    Uniform Convergence of Sequences

    Homework Statement For each of the following sequences (fn), find the function f such that fn --> f. Also state whether the convergence is uniform or not and give a reason for your answer. Homework Equations a.) fn(x) = 1/xn for x greater than or equal to 1 b.) f[SUB]n[SUB](x) =...
  25. J

    Dirichlet test for convergence

    Homework Statement use dirichlet test to determine if the series converges 1-1/2-1/3+1/4+1/5-1/6-1/7+... Homework Equations The Attempt at a Solution I have broken up the series into two different series the first series I have is 1+1/4+1/8+1/12+... and the second series I...
  26. S

    Why does the ln(x) series converge for x=2 but not x=0?

    Homework Statement This isn't really so much a homework problem as me asking a question. The Taylor Series for ln(x) centered at 1 is: sum_[0, infinity] ((-1)^((n+1)*(x-1)^n))/n, then why does it converge for the endpoint x=2, but not x=0 of the interval of convergence? Homework Equations The...
  27. B

    Uniform convergence ( understanding how to apply a theorem)

    Homework Statement show a function f_n is not uniformly convergent using a theorem: Homework Equations if f_n converges uniformly to F on D and if each f_n is cont. on D, then F is cont. on D The Attempt at a Solution not really sure what to do. use the contrapositive? would that...
  28. Telemachus

    Radius of Convergence for Power Series: What is the Limiting Ratio Test?

    Homework Statement Hi there. Well, I was trying to determine the radius and interval of convergence for this power series: \displaystyle\sum_{0}^{\infty} \displaystyle\frac{x^n}{n-2} So this is what I did till now: \displaystyle\lim_{n \to{+}\infty}{\left...
  29. C

    Convergence of an Integral

    Hi, I need to determine whether this improper integral converges or diverges \int_{-1}^{1} \frac{x}{\sqrt{1-x^2}}dx The original function DNE at -1, 1 so I split the limits \int_{-1}^{0} \frac{x}{\sqrt{1-x^2}}dx \ + \ \int_{0}^{1} \frac{x}{\sqrt{1-x^2}}dx I've...
  30. H

    Convergence of subseries of the harmonic series

    I need to show that the by eliminating infinitely many terms of the harmonic series, the remaining subseries can be made to converge to any positive real numbers. I have no clue to prove this. I know harmonic series diverges really slowly, will this fact come into play? Thank you very much!
  31. F

    The proof of convergence. I am confused with the summation

    Homework Statement Prove\; that\;if\;\sum_{n=1}^{\infty} a_n \;converges,\;then \lim_{n\to\infty}a_n = 0 Book solution s_n= a_1 + a_2 +...+a_n s_{n-1}= a_1 + a_2 +...+a_{n-1} a_n=s_n-s_{n-1} Then they did a few limits, and proved that the difference is 0. BUt that is not my...
  32. E

    Convergence of an infinite series

    Homework Statement http://img840.imageshack.us/img840/3609/unleddn.png note that by log(n), i really mean NATURAL log of n Homework Equations it's convergent, but I can't figure out which test to useThe Attempt at a Solution there is no term to the nth power, so ratio test is useless; root...
  33. L

    Determining the interval of convergence

    Homework Statement f(x)=x^{0.4} Construct a power series to represent the function and determine the first few coefficients. Then determine the interval of convergence. The Attempt at a Solution Determining the first few coefficients is simple enough. Take the first few...
  34. M

    Uniform Convergence of g_n (x): Proof & Subset Analysis

    1. Let g_n (x)=nx*exp(-nx). Is the convergence uniform on [0, ∞)? On what subsets of [0,∞) is the convergence uniform? 3. I am looking for a proof of how the convergence is uniform (possibly using Weierstrass' M Test?). I understand that the subset that determines uniform convergence is...
  35. D

    Uniform vs pointwise convergence

    I was reading Royden when I came across this cryptic statement:pg 222, "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise convergence is not a metric concept." Can anyone illuminate this?
  36. I

    Convergence of the Arctangent Integral

    Homework Statement Is $\intop_{-\infty}^{\infty}\arctan(x)\, dx$ convergent? What about $\lim_{t\rightarrow\infty}\intop_{-t}^{t}\arctan(x)\, dx$?Homework Equations The Attempt at a Solution I think the first integral may actually be divergent the way its written and the second one...
  37. S

    Finding radius of convergence of series ?

    Homework Statement How would I find the radius of convergence of this series? f(x)=10/(1-3x)2 is represented as a power series f(x)=\sum from n=0 to \infty CnXn Homework Equations The Attempt at a Solution Okay so I tried deriving, using d/dx(1/1-3x)=3/(1-3x)2 and ended up with...
  38. P

    Abs. conv, convergence, or divergence

    Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \sum (-1)^n\frac{e^{1/n}}{n^4} Homework Equations The Attempt at a Solution I used the root test so \sqrt[n]{\frac{e^{1/n}}{n^4}} --> \lim_{n\to \infty...
  39. K

    Interval of Convergence

    Homework Statement The summation from n=1 to infinity of ((n!)x^(2n))/((2n-1)!) Find the Interval of Convergence of this series. Homework Equations Ratio test The Attempt at a Solution I applied the ratio test, then got x^2 times the limit as n approaches infinity of...
  40. S

    Finding the radius of convergence and interval of convergence

    Homework Statement This is the question of mine that I'm having a little confusion about. I know the whole process in which you use the ratio test to determine the radius of convergence and using that you test the end points of the summation to see if they converge at the end points aswell...
  41. A

    Difficult(?) convergence problem

    Homework Statement Show that if \vartheta is any constant not equal to 0 or a multiple of 2\pi, and if u_{0}, u_{1}, u_{2} is a series that converges monotonically to 0, then the series \sum u_{n} cos(n\vartheta +a) is also convergent, where a is an arbitrary constant. Homework...
  42. K

    Determine Convergence or Divergence

    Homework Statement Determine if the series the summation form n=2 to infinity of n/((n2+1)ln(n2+1)) is convergent or divergent. Homework Equations The Attempt at a Solution I applied the integral test and got positive infinity, so I it diverges. But I want to know if I'm right...
  43. M

    Radius/Interval Convergence for Power Series

    Homework Statement Find the radius and interval of convergence for the power series of n=0 of infinity of n^3(x-5)^n Homework Equations Ratio test: http://en.wikipedia.org/wiki/Ratio_test The Attempt at a Solution [(n+1)^3(x-5)^n+1 / n^3(x-5)^n] I am lost as to how to...
  44. T

    Convergence of a series by root test

    Homework Statement Does the series: sum from 1 to inf. (n+2)/(n+1) converge? If so does it converge absolutely? Homework Equations Ratio test for series The Attempt at a Solution I found this series to converge using the root test, (.jpg attached) however wolfram alpha...
  45. M

    Convergence of ∫dx/sqrt(x^4+1): Explanation Needed

    Homework Statement Does ∫dx/sqrt(x^4+1) from x=-∞ to x=∞ converge or diverge? explain in detail if you can please. thanks Homework Equations limit comparison test direct comparison The Attempt at a Solution...well i have the answer, it converges. I just need a better...
  46. M

    How can I efficiently test for convergence in integrals without wasting time?

    Homework Statement Hi, So i don't know if this is a stupid question but i'll ask anyways. So I'm on the chapter where we start testing integrals for convergence. The books starts out with elementary functions then they move towards non elementary functions. Testing for them is OK, my problem...
  47. A

    Can someone please confirm or deny this assertion about absolute convergence?

    EDIT: On pg. 390 of Kreyszig's Functional Analysis text, we have: "If T is a bounded linear operator on a nonempty Banach space, then the series \sum_{k=0}^\infty \left( \frac{T}{\lambda} \right)^k converges absolutely for |\lambda| > 2\| T \|." The argument presented in Kreyszig...
  48. M

    Power Series - Interval of Convergence Problem

    Homework Statement For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.) Summation of n=1 to infinity of (n!)^2 / (kn)! Homework Equations ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the...
  49. B

    Finding Integral Convergence Points

    Homework Statement I'm tasked with integrating the following functions, and values for t where the function converges: \int_{0}^{1}x^p\cdot ln(x) dxHomework Equations Integration by Parts Formula: \int udv=uv-\int vdu The Attempt at a Solution I found a definite integral...
  50. R

    Interval of Convergence: Series f(x)

    Homework Statement A function f is defined by... f(x) = \frac{n+1}{3^{n+1}} x^n a.) find the interval of convergence of the given power series. b.) Find \lim_{x\rightarrow 0} \frac{f(x) - \frac{1}{3}}{x} c.) Write the first three nonzero terms and the general term for...
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