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

    Help Convergence of Power Series, interval and radius of convergence question

    Homework Statement Determine the radius of convergence, the interval of convergence, and the sum of the series Summation from k=2 to ∞ of k(x-2)^k+1. Homework Equations ratio test? The Attempt at a Solution possibly take the derrivitive of the power series, then find the sum then integrate...
  2. E

    Finding Domain of Convergence for Complex Series

    Homework Statement I need to find the domain of absolute convergence of the following series: ^{\infty}_{1}\sum(z+3)^{2n}/(2n)! Homework Equations Ratio test? The Attempt at a Solution I'm not really sure how to handle the complex variable z within the series. I attempted to use the...
  3. H

    Is superlinear convergence always better than linear convergence?

    Homework Statement http://im2.gulfup.com/2011-04-01/1301686351321.gif Homework Equations superlinearly convergence The Attempt at a Solution [PLAIN][PLAIN]http://im2.gulfup.com/2011-04-01/1301686616101.gif this is what i know about it, kindly help me
  4. S

    Power Series: Find Interval & Radius of Convergence

    Homework Statement \Sigma (from index k = 1 until infinity) Within the Sigma is the series : (k! * (x^k)) Homework Equations Ratio Test : lim as k approaches infinity |a(k+1) / ak| The Attempt at a Solution When I apply the ration test to the series and simplify I get lim k...
  5. T

    Convergence of a Taylor Series

    Homework Statement Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges? Note: 'sum' represents the sum from n=0 to infinity Homework Equations tanh(z) =...
  6. I

    Does the integral of x^(3/2) sin(2x) converge over the range of 0 to 1?

    Homework Statement Discuss the convergence of Integral(x3/2 sin 2x dx) range is from 0 to 1Homework Equations|sin x| =< 1 The Attempt at a Solution The sine function converges absolutely. It is also increasing from 0 to 90 degrees, decreases until 27 decrease and it is negative from 180 to...
  7. S

    Series Convergence: Does \sum_{k=1}^{\infty }{a_k}^{5/4} Necessarily Converge?

    Homework Statement Say that \sum_{k=1}^{\infty }a_k converges and has positive terms. Does the following necessarily converge? \sum_{k=1}^{\infty }{a_k}^{5/4} Homework Equations If it necessarily converges, a proof is required, if not, a counter-example is required. The...
  8. S

    Basic Complex Analysis: Uniform convergence of derivatives to 0

    Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...
  9. H

    Convergent Sum of Sine Series with Added Term for Continuity

    Find the convergent sum and find the sum of first five terms \sum_{n=1}^{\infty} \frac{sin(nx)}{2^nn} from 1 to infinity. I have found so far that: \sum_{n=1}^{\infty} \frac{sin(nx)}{2^n} = \frac{2sin(x)}{5-4cos(x)} I am not sure how to consider the \frac{1}{n} term. Can someone please help?
  10. R

    Convergence Tests: 3 Problems Explained

    Homework Statement Determine if the following converge: a. (∞,n=1) ∑ (1+1/n)^n b. (∞,n=1) ∑ sin(n)/(n^2 + √n) c. (∞,n=3) ∑ 1/(k ln^2 k) The Attempt at a Solution a. I tried the root test, but it failed, so i immediately went to the limit divergence test...ended up getting...
  11. L

    Convergence of a geometric series; rewriting a series in the form ar^(n-1)

    Homework Statement Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \sumn=1infinity (-3)n-1/4nHomework Equations A geometric series, \sumn=1infinity arn-1=a + ar + ar2 + ... is convergent if |r|< 1 and its sum is \sumn=1infinity arn-1 =...
  12. L

    Determine the convergence of the sequence e^(1/n).

    Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. an = e1/n Homework Equations The limit laws, adapted for sequences. The Attempt at a Solution I have the solution; I was just wondering if someone might explain it to me. I...
  13. H

    Question: Series Convergence for ((-1)^n*n!)/(1*6*11*...*(5n+1))

    Homework Statement Does the series ((-1)^n*n!)/(1*6*11*...*(5n+1)) from n = 0 to \infty absolutely converge, converge conditionally or diverge? Homework Equations The Attempt at a Solution I did the ratio test for ((-1)^n *n!)/(5n+1)) and I found that it diverges but apparently...
  14. L

    Determining the absolute convergence, convergence, or divergence of a series.

    Homework Statement \Sigma from n=0 to infinity (-10)n/n! Determine the absolute convergence, convergence, or divergence of the series. Homework Equations In this section, it's suggested that we use the following to determine a solution: A series is called absolutely convergent if the series...
  15. C

    Finding the Radius & Interval of Convergence of a Power Series

    Homework Statement \sum from n=1 to inf (1+ 1/2 + ... 1/n)x^n Find the radius of convergence and the interval of convergence of the given power series. Homework Equations Dunno.. The Attempt at a Solution Stuck thinking about it. I'm not sure if I can combine what's in brackets with the...
  16. N

    Convergence of an improper integral

    Homework Statement For what values of r does \int(from 0 to infinity) xre-x dx converge? I assume that the problem refers to r as any real number. 2. The attempt at a solution I have given this a try but I am really not confident that I did it right... First i used integration...
  17. M

    Distribution Theory - Uniform Convergence

    F: C(Omega) -> D'(Omega); F(f) = F_f -- O = Omega Introduce the notion of convergence on C(Omega) by f_p -> f as p -> inf in C(O) if f_p(x) -> f(x) for any xEO Show that then F is a continuous map from C(O) to D'(O) Hint: Use that if a sequence of continuous functions converges to a...
  18. E

    Real Analysis convergence proof

    Homework Statement If the sequence xn ->a , and the sequence yn -> b , then xn - yn -> a - b The Attempt at a Solution Can someone check this proof? I'm aware you cannot subtract inequalities, but I tried to get around that where I indicated with the ** in the following proof...
  19. C

    Convergence Tests: Test for Convergence of Series

    Homework Statement Test for convergence \sum sin(1/n) / \sqrt{ln(n)} sum from 2 to inf Homework Equations Limit comparison test if lim an / bn to inf doesn't equal 1 you know if it converges or diverges by limit comparison test The Attempt at a Solution I've tried a lot of different...
  20. J

    Pointwise and uniform convergence of sequence of functions

    Homework Statement show if has a uniform convergence of pointwise also we know that x gets values from 0 to 1The Attempt at a Solution for the pointwise I think its easy to show that limfn(x) as n->infinity is 0 but I am really stuck in uniform convergence I know that fn converges...
  21. A

    L^2 convergence implies a.e. pointwise convergence? Since when?

    My functional analysis professor made the following assertion the other day: If f_n \to f in the L^2 norm, then there is a subsequence f_{n_k} that converges pointwise almost everywhere to f. This is the first I've heard of that...can someone point me to a proof of this proposition? Does it have...
  22. B

    Does the series converge to pi^2/8?

    Homework Statement Homework Equations [PLAIN]http://img577.imageshack.us/img577/6756/physforumsquestion.png The Attempt at a Solution I am pretty much stuck guys. Any help will be greatly appreciated!
  23. T

    Topology - Use Componentwise Convergence Criterion to prove closed ball closed.

    Homework Statement Let r be a positive number and define F = {u in R^n | ||u|| <= r}. Use the Componentwise Convergence Criterion to prove F is closed.Homework Equations The Componentwise Convergence Criterion states: If {uk} in F converges to c, then pi(uk) converges to pi(c). That is, the...
  24. S

    Proving Integral Convergence with L1 Functions

    Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given. Given: Suppose f is contained in L1([a,b]) Prove for almost everywhere x is contained in [a,b] limit as h goes to 0+, int (abs(f(x+t)+f(x-t)-2f(x)))dt = 0 Initially I...
  25. G

    Series (Convergence, determination, and error)

    Homework Statement Approximate the sum of the series S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n!} by calculating S_10. Estimate the level of error involved in this problem. AND S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n^4} Approximate the sum of the series by...
  26. H

    Stone-Weierstrass, uniform convergence

    Homework Statement Show that there are continuous functions g:[-1,1]\to R such that no sequence of polynomials Q_n satisfies Q_n(x^2)\to g(x) uniformly on [-1,1] as n\to\infty The Attempt at a Solution Suppose there is a sequence Q_n such that Q_n(x^2)\to g(x) uniformly for g(x)=x. Then...
  27. T

    What Determines the Radius of Convergence in Complex Power Series?

    Homework Statement [PLAIN]http://img153.imageshack.us/img153/4822/radiusm.jpg Homework Equations The Attempt at a Solution Using the ratio test: \left | \frac{e^{i(n+1)^2 \theta} \theta^{n+1} z^{(n+1)^2}}{e^{in^2 \theta} \theta ^n z^{n^2}} \right | = | \theta...
  28. T

    Absolute Convergence: Complex Homework Equations and Solutions

    Homework Statement [PLAIN]http://img225.imageshack.us/img225/7501/complexh.jpg Homework Equations The Attempt at a Solution How do I show this absolutely converges?
  29. P

    Help Convergence test for series

    Find the value of x so that the series below converge.[/b] \sum 1/ [(k^x) * (2^k)] (k=1 to \infty) Using ratio test, I 've got [(1/2) * 1^x] < 1 for all x in R But when I use different value of x, series converge and diverge! Really need your help! Thank you so much
  30. estro

    Proving Sequence Convergence: Tips & Tricks

    I know that the sequences meets the following: (n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n) I've got the feeling that this sequence is alternating or decreasing, but I was unable to prove it. Usually I use induction to prove things about such inductive sequence but in this case I don't have real values t o...
  31. estro

    Dirichlet's Convergence Test - Improper Integrals

    Hello, I have question about using Dirichlet's Convergence Test which states: 1. if f(x) is monotonic decreasing and \lim_{x\rightarrow \infty} f(x)=0 2. G(x)=\int_a^x g(t)dt is bounded. Then \int_a^\infty f(x)g(x)dx is convergent. But what about the following situation: f(x)=1/x g(x)=cosxsinx...
  32. T

    Convergence Laws in Probability

    Homework Statement Hello, I'm trying to revise for my probability exam next week and am getting a bit hung up on how to show a sequence of random variables converges. For example, how would I got about doing this question: Let {X_{n}:n\geq1} be a sequence of random variables with...
  33. C

    Interval of Convergence for a Series

    Homework Statement Ok, so I don't need help with this part, I just got stuck at the following step when attempting to find the interval of convergence: The Attempt at a Solution I got here: -4 < x^2 < 4 So, I need to solve this inequality. But can I? How can I take the square root...
  34. C

    Convergence in the product and box topology

    Hi. Can I have some help in answering the following questions? Thank you. Let {f_n} be a sequence of functions from N(set of natural numbers) to R(real nos.) where f_n (s)=1/n if 1<=s<=n f_n (s)=0 if s>n. Define f:N to R by f(s)=0 for every s>=1...
  35. D

    The rate of convergence of a sequence

    It's been a while since I've done rate of convergence problems, how would i find the rate of convergence for either of these sequences? 1) limn->infsin(1/n)=0 2)limn->infsin(1/n^2)=0
  36. O

    Proof About Series Convergence

    Homework Statement let an be a positive series. it is known that for every bn\rightarrow \infty the sum from 1 to inf of an/bn is convergent prove that the sum from 1 to inf of an is convergent Homework Equations The Attempt at a Solution I thoght maybe to try to say. let...
  37. P

    Convergence of Series: Homework Statement and Attempt at Solution

    it's a new semester, and we're at it again. series abounds! Homework Statement Let . Determine whether the following series converges: Homework Equations -1 \leq \sigmak \leq 1 The Attempt at a Solution i feel like the series inherently diverges because of the 1/k element...
  38. G

    Complex Analysis: Radius of Convergence

    Homework Statement Find the radius of convergence of the power series: a) \sum z^{n!} n=0 to infinity b) \sum (n+2^{n})z^{n} n=0 to infinity Homework Equations Radius = 1/(limsup n=>infinity |cn|^1/n) The Attempt at a Solution a) Is cn in this case just 1? And plugging it in...
  39. D

    Ratio test for convergence of series 1/log(n)?

    Hi. I have been banging my head against this problem for a while and I just don't get it. Maybe (probably) it's something wrong with my logarithm-fu or limit-fu. I just registered to ask this because I couldn't find an answer anywhere, and I've been reading these forums for a while for other...
  40. E

    A very tough convergence (-1)^n * [e-(1+1/n)^n]

    Hi and sorry if I misplaced the thread. I'm having quite some trouble with analyzing the convergence of the following series : Homework Statement :[/B] Determine whether the series is convergent or divergent, absolutely & normally. \sum (-1)^n * [e-(1+1/n)^n] Homework Equations...
  41. E

    A very tough convergence: (-1)^n * [e-(1+1/n)^n]

    Hi and sorry if I misplaced the thread. I'm having quite some trouble with analyzing the convergence of the following series : \sum (-1)^n * [e-(1+1/n)^n] I had troubles both with absolute and normal convergence. With normal convergence I tried Leibniz 1) lim a(n) = 0 Which is ok { lim...
  42. U

    Rate of convergence for functions

    I am not very familiar with terms from numerical analysis, thus I do understand the definition for convergence rate from http://en.wikipedia.org/wiki/Rate_of_convergence" . Still, here the definition appears only for sequences. Which is the definition for rate of convergence for functions? For...
  43. H

    Power Series Interval of Convergence

    Homework Statement I need a power series with a radius = pi. (So when you do the ratio test on this power series you get pi) Homework Equations The Attempt at a Solution I tried x^n*sin(n) and thought of stuff like that but couldn't come up with a working power series
  44. Z

    A question about convergence with probability one

    Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this? Thanks a lot.
  45. H

    Series Absolute Convergence Proof

    Homework Statement If \suman2 and \sumbn2 converge show that \sumanbn is absolutely convergent Homework Equations The Attempt at a Solution I think I should do something with the statement 2ab\leq a^2 + b^2
  46. H

    Proving Convergence of a Series

    Homework Statement (\sqrt{(n+1)} - \sqrt{n} ) / \sqrt{n} I'm trying to show this series converges.Homework Equations divergence test: as n approaches infinity, if the sequence does not approach 0 then the series diverges ratio test: as n approaches infinity, if the ratio between subsequent...
  47. maverick_starstrider

    Understanding Convergence Factors in Physics Textbooks

    "Convergence Factors" In all my textbooks I always see these random convergence factors thrown in (+0's or +i*nu or some such) but I have never seen a book that would dirty itself by steeping so low as to explain what they are (I'm looking at you Wen, Bruus and Flensberg, Fetter and Walecka...
  48. chwala

    Convergence of ode & pde

    Hi, well let me put the question a bit clear...my concern area is on ode and pde...my question is when you solve a pde/ode analytically and get a solution by asymptotic means does this mean that if solution exists then ...when using convergence as an alternative way of getting solution of the...
  49. chwala

    Does convergence imply asymptotic relation

    hello, does convergence imply asymptotic relation of an ordinary differential equation?
  50. A

    Need a proof about convergence of a series

    I was reading a book about Calculus that I came to a problem that the author claimed convergence of a series won't change if we subtract a finite number of its terms from it, It seems to be intuitively clear, but I need a proof. so please Prove that the convergence/divergence status of a series...
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