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. R

    Explore the Convergence of Drug Levels with Math, Physics, Biology, & Chemistry

    Okay, this question straddles math, physics, biology, and chemistry, so I'm not sure if this is the correct forum to post it in, but I was using mostly chemistry knowledge to solve it, so I would guess that may be the correct method. Anyway, it's not homework, but it is similar to many homework...
  2. C

    Convergence of Improper Integrals: Exploring the Power of p

    I'm trying to refresh my knowledge of Calc II, and I'm going through improper integrals right now. The problem I am trying to solve is: For which numbers p\geq0 does \int_0^\infty \frac{e^{-x}}{x^{p}} converge? Justify your answer. So far, I've split up the integral into two halves (0 to 1...
  3. A

    Convergence of Bisection Method

    Homework Statement Show that the Bisection Method converges linearly with K = 1/2 Homework Equations Note that x(sub n) converges to the exact root r with an order of convergence p if: lim(n->oo) (|r - x(n + 1)|) / (|r - x(n)|^p) = lim(n->oo) (|e(n + 1)|) / (|e(n)|^p) = K The...
  4. G

    Question on Radius of Convergence for values of x, when f(x) is x^2

    Homework Statement This is not so much an entire problem I need help with but just a part. It is a power series where after you do the ratio test, you end up with |4x^(2)| < 1, so |x^(2)| < 1/4. Since the radius of convergence is |x-a| < R, I end up with -1/4 < x^(2) < 1/4, but because...
  5. K

    The Convergence of Linguistics and Mathematics

    I have a question that I've been pondering recently. As far as I can tell, it's original to the boards or at least hasn't been discussed in a long time so I think it's fair to start a new topic. This concerns initial bases for thought. It would seem that both language and mathematics are the...
  6. M

    Find the Summation Notation and the Radius of Convergence

    Homework Statement Find the Summation Notation and Radius of Convergence of this series. 5, x, 10, x, ... The Attempt at a Solution I don't know how did they come up with that equation.. But the summation seems right.. Can anyone tell me how did they arrive with that equation? I've tried...
  7. Z

    Will This Mathematical Series Converge?

    Homework Statement Test for convergence or divergence. Give a reason for your decision. Homework Equations \sum_{i=1}^{\infty} \frac{\sqrt{2n-1} \log{(4n + 1)}}{n(n + 1)} The Attempt at a Solution I've tried to compare it to the series \sum_{i=1}^{\infty} \frac{\sqrt{2n-1} \log{(4n +...
  8. L

    Uniform convergence, mean convergence, mean-square convergence

    Hi, I was always troubled by the relationships between these modes of convergence (L^1, L^2, and L^{\infty} convergences, to be precise), so I took some books and decided to establish some relations between them. For some, I succeeded, for others I did not. Here's what I did so far: If I...
  9. icesalmon

    Absolute or Conditional Convergence, or Divergence of Alternating Series.

    Homework Statement given an = ( -1 )(n+1) / \sqrt{n} determine if the infinite series is Absolutely Convergent, Conditionally Convergent, or Divergent. Homework Equations I hope I have these theorems down correctly, please correct me if I'm wrong. If \Sigma|an| is Convergent then...
  10. E

    Does the Series Involving a Function with Continuous Third Derivative Converge?

    This problem has been bothering me for some time. Any thoughts or insights are greatly appreciated. Consider a function, f, with continuous third derivative on [-1,1]. Prove that the series \sum^{\infty}_{n=1} (nf(\frac{1}{n})-nf(-\frac{1}{n}) - 2\frac{df}{dn}(0)) converges. Thanks in...
  11. M

    How to determine convergence and divergence

    I've been having some trouble understanding how to determine if a sequence is divergent or convergent. For example an = cos(2/n) I know if I take the limit as n ->\infty then I will get 1. So the sequence has a limit but does having a limit mean that the sequence is convergent.
  12. 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...
  13. 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...
  14. 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}...
  15. 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...
  16. 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...
  17. 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...
  18. R

    Does Convergence in Distribution Guarantee Probability Equality for Events?

    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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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 }...
  29. dextercioby

    What Is an Example of Weak but Not Strong Convergence in L²(R)?

    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...
  30. 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...
  31. 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...
  32. 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!
  33. 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...
  34. 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...
  35. 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) =...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. C

    Does This Improper Integral Converge or Diverge?

    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...
  41. 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!
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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?
  47. 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...
  48. 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...
  49. 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...
  50. K

    What Is the Interval of Convergence for the Given Series?

    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...
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