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. Mr. Rho

    How to show this convergence?

    Homework Statement I need to show that \sum\limits_{n=0}^\infty \frac{sin^{4}(\frac{n\pi}{4})}{n^2} = \frac{\pi^{2}}{16} Homework Equations I have this property for odd n \sum\limits_{n=0}^\infty \frac{1}{n^2} = \frac{\pi^{2}}{8} The Attempt at a Solution [/B] I have no idea how to do...
  2. C

    Convergence or Divergence of Factorial Series

    How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...
  3. C

    Convergence or Divergence of ∑ tan(1/k) for k=5

    Homework Statement ∞ ∑ tan(1/k) k=5 show that it is convergent or divergent Homework EquationsThe Attempt at a Solution i used ratio test, but it's equal to 1, it means no works... i used divergence test, it equals to 0, no work too... so what should i do? i don't know how to use...
  4. titasB

    Infinite Series Convergence using Comparison Test

    Homework Statement Determine whether the series is converging or diverging Homework Equations ∞ ∑ 1 / (3n +cos2(n)) n=1The Attempt at a Solution I used The Comparison Test, I'm just not sure I'm right. Here's what I've got: The dominant term in the denominator is is 3n and cos2(n)...
  5. M

    Finding Taylor Series for Exponential Functions

    Hello, For the exercises in my textbook the directions state: "Use power series operations to find the Taylor series at x=0 for the functions..." But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n. Here...
  6. Destroxia

    Convergence of 1/K?: Tests & Solutions

    Homework Statement Does (1/(k!)) converge? Homework Equations [/B] Convergence Tests?The Attempt at a Solution I thought I could just simply use the divergence test, but I'm not sure if that only tells you if it's divergent and not whether it is convergent or not. lim(k>inf) (1/(k!)) = 0...
  7. C

    Quick question about Ratio Test for Series Convergence

    Homework Statement [/B] This is the question I have (from a worksheet that is a practice for a quiz). Its a conceptual question (I guess). I understand how to solve ratio test problems. "Is this test only sufficient, or is it an exact criterion for convergence?" Homework Equations Recall the...
  8. E

    Convergence problem in ANSYS Workbench

    Hi, i solved a model but i got an error saying that "The solver engine was unable to converge on a solution for the non linear problem as constrained". Can someone offer advice on how i can fix this error?
  9. M

    MHB Integral: Investigating Convergence II

    Investigate the convergence of the integral $\int_{0}^{\infty} \frac{x^{ \frac{2}{3}} + \frac{2}{3} \ln x }{1+x^2} \mbox{d}x$
  10. M

    MHB Integral: Investigating Convergence I

    Investigate the convergence of the integral $\int_{0}^{\infty} \frac{-\ln \left[ \frac{1}{a} x ^{ \frac{2}{a-1} } e ^{- x^{\frac{2}{a} }} \right] }{1+x^2} \mbox{d}x$ for $a \ge 2$
  11. K

    MHB Uniform convergence of a complex power series on a compact set

    I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...
  12. N

    Complex Analysis: Series Convergence

    Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...
  13. P

    Regulated Integral with Convergence Factor

    I have a question pertaining to a computation I'm trying to carry out. Without getting too much into the details, I have a finite integral over two variables. One integral vanishes, and one diverges allowing for the finite value. I had to regulate the divergent integral so I introduced a...
  14. P

    Real analysis, sequence of sequences convergence proof

    Homework Statement \ell is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is d(x,y) = sup|x_n - y_n|, for all n in naural-numbers then the sequence u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...} and...
  15. C

    Convergence or Divergence of a series

    Homework Statement Does sum from n=1 to n=infinity of 1/[n^(1+1/n)] converge or diverge. Homework Equations ^^^^^^^^^^^^^^^ The Attempt at a Solution The general term goes to 0 and its a p-series with p>1, but for large n the series becomes 1/n pretty much so, even tho p>1 is it divergent?
  16. C

    Proof of convergence and divergence

    Homework Statement Does \frac{2^{n}}{n!} converge or diverge? The Attempt at a Solution Is there more than one way to prove this? I would appreciate a few directions. I've been trying the Squeeze theorem for a long time. I said 1/n! was smaller, but I have no damn idea how to say what's...
  17. J

    Fixed point iteration, locally convergent

    Homework Statement For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.) A) 1/x2 -1 B)... C)... compute xbar to within absolute error 10-4. Homework Equations 3. The...
  18. Shackleford

    What is the radius of convergence of

    Homework Statement z ∈ ℂ What is the radius of convergence of (n=0 to ∞) Σ anzn? Homework Equations I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences. a_n = \frac{n+(-1)^n}{n^2} limn→∞ |an|1/n The Attempt at a Solution I think that the radius of...
  19. M

    Convergence of infinite series (e^(1/n)-1)

    Homework Statement Part a.) For a>0 Determine Limn→∞(a1/n-1) Part b.) Now assume a>1 Establish that Σn=1∞(a1/n-1) converges if and only if Σn=1∞(e1/n-1) converges. Part c.) Determine by means of the integral test whether Σn=1∞(e1/n-1) converges Homework Equations Integral Test Limit...
  20. N

    ChetIs the Integral Convergent?

    Homework Statement Find whether the integral is convergent or not, and evaluate if convergent. Homework Equations integral 1/sqrt(x^4+x^2+1) from 1 to infinity The Attempt at a Solution 1/sqrt(x^4+x^2+1)<1/sqrt(x^4) 1/sqrt(x^4)=1/x^2 which is convergent for 1 to infinity and is 1 therefore...
  21. Coffee_

    QM, the convergence of the harmonic oscillator function.

    1. After finding out that the wave function ##\Psi(z) \sim Ae^{\frac{-z^{2}}{2}}## in the limit of plus or minus infinity Griffiths separates the function into two parts ##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}## My question will be about a certain aspect of the function ##h(z)## After solving the...
  22. Ignis Radiis

    Series Convergence and Divergence test

    Homework Statement So my question was Sum- (n=2) ln(n)/n Homework Equations I noticed that you can only limit comparison, because so far, I have tried doing all the other test such as the nth term test, p-series, integral(i have no idea how to integrate that). The Attempt at a Solution
  23. B

    Uniform Convergence: Understanding the Limitations and Implications

    So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence " (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) " My problem is that isn't...
  24. M

    Why is the commutative property not true for conditionally convergent series?

    I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important? I have not really experienced a time where I am rearranging terms...
  25. B

    Rigid Body Motion/Out of convergence in ansys Classic 14.5

    Hi all, I am analysing a 2D D-shaped Neo Hookean model in contact with a Rigid link. The details are in the input file attached. Can someone guide me on how to solve this issue?:headbang: Thanks, Bruce
  26. G

    Limit for a problem of convergence

    Homework Statement I ultimately want to discuss convergence of the integral \int_{0}^{\infty}\frac{1}{\sqrt{x}e^{\sqrt{x}}}dx[/B]Homework Equations \int_{c}^{\infty}\frac{dx}{x^{p}} is convergent near x approaching infinity for p>1 3. The Attempt at a Solution While I understand that the...
  27. C

    Proof of Convergence: ∑∞n=1 n/(3n + n2)

    Homework Statement I have been asked to prove the convergence or otherwise of ∑∞n=1 n/(3n + n2). In the example solution, with the aim to prove divergence by comparison with the Harmonic Series, the lecturer has stated that n/(3n + n2) ≥ n/(4n2) = 1/4n and which diverges to +∞. I was...
  28. ironman

    Solving Limits: Find Interval & Radius of Convergence

    Homework Statement [/B] I have to find the radius of convergence and convergence interval. So for what x's the series converge. The answer is supposed to be for every real number. So the interval is: (-∞, ∞). So that must mean that the limit L = 0. So the radius of convergence [ which is...
  29. P

    Proving convergence of integral

    Homework Statement Prove the following double integral is convergent. ##\int_0^1 \int_0^1 \frac{1}{1-xy}\, dx \, dy## The Attempt at a Solution This was a bonus question on my final exam in calc 3 yesterday, I just want to show my steps and see if they were right. So I realized that...
  30. nuuskur

    Is this series conditionally convergent?

    Homework Statement Consider the series: \sum\limits_{k=17}^\infty (-1)^{k}(\sqrt{k-3}-\sqrt{k-5}) Homework EquationsThe Attempt at a Solution First, I will attempt to determine whether it is absolutely convergent: \lim\limits_{k\to\infty} \left(\sqrt{k-3}-\sqrt{k-5}\right) = 0 Since the limit...
  31. C

    Another Laplace Transform problem, need region of convergence help

    Homework Statement Find L[x(t)], where $$ x(t) = tu(t) + 3e^{-1}u(-t) $$ Also determine the region of convergenceHomework EquationsLaplace properties, Laplace table: L[te-at = 1/(s+a)2 L[u(t)] = 1/s L[t] = 1/s2 The Attempt at a Solution I don't really know what to do with this as my table...
  32. G

    Finding Convergence Radius & Interval: Solving a Complex Homework Problem

    Homework Statement ∞ n=3 ∑ ((-1)n (x+3)3n)/(2nlnn) Find radius of convergence, interval of convergence, values for x which series is: absolutely convergent, conditionally converge or divergence. Homework EquationsThe Attempt at a Solution I applied the Ratio Test and got |(x+3)3| lim...
  33. M

    MHB Convergence as for the norm

    Hey! :o If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to show it?? (Wondering) What does convergence as for the norm mean?? (Wondering)
  34. C

    Ratio Test Radius of Convergence

    Homework Statement ∑ x2n / n! The limits of the sum go from n = 0 to n = infinity Homework EquationsThe Attempt at a Solution So I take the limit as n approaches infinity of aa+1 / an. So that gives me: ((x2n+2) * (n!)) / ((x2n) * (n + 1)!) Canceling everything out gives me x2 / (n + 1)...
  35. M

    Radius of convergence derivation

    Hi, I am likely just missing something fundamental here, but I recently just revisited series and am looking over some notes. In my notes, I have written that if ## \lim_{x \to +\infty} \frac{a_{n+1}}{a_n} = L ## Then ## | x - x_o | = 1/L ## But shouldn't the correct expression be $$ | x -...
  36. R

    MHB Pointwise convergence of holomorphic functions

    Hello. In my complex analysis book I've read a theorem which says that if a sequence \{ f_n \} of holomorphic functions on a domain \Omega converges pointwise to a function f, then f is holomorphic on a dense, open subset of \Omega. I know how to prove this theorem. I just find it hard to...
  37. R

    Laplace transform and region of convergence

    Find the LT and specify ROC of: x(t) = e-at, 0 ≤ t ≤ T = 0, elsewhere where a > 0 Attempt: X(s) = - 1/(s+a)*e-(s+a) integrated from 0 to T => -1/(s+a)[e-(s+a) + 1] Converges to X(s) = 1/(s+a) , a ⊂ R, if Re{s} > -a for 0≤t≤T Elsewhere ROC is empty (LT doesn't exist). Is this...
  38. G

    Sequence (n)/(n^n) Convergent or Divergent and Limit?

    Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...
  39. L

    Proving Convergence of an = [sin(n)]/n w/ Cauchy Theorem

    Homework Statement an = [sin(n)]/n Prove that this sequence converges using Cauchy theorem Homework Equations Cauchy theorem states that: A sequence is called a Cauchy theorem if for all ε > 0, there exists N , for all n > N s.t. |xn+1 - xn| < εI do not know how to approach this proof. I...
  40. P

    Convergence Doubt: Is Most Repeated Value the Answer?

    Hello people, In case if I am typing the question in the wrong forum please redirect me. OK, here it goes... I have this stupid question: Suppose we have a sequence 1, 1, 1, 1, 1, 1... It converges to '1'. Consider 1, 0, 1, 0, 1, 0... it diverges right? What about a sequence 1, 1, 0, 1, 1...
  41. N

    Question: Super Convergence and Convergence: Is There a Relationship?

    Homework Statement Suppose (## a_n ##) is a sequence and let l\in\mathbb R. Let us say that (## a_n ##) is "super convergent" to ##l## if there exists N\in\mathbb N such that for every ε>0 we have ##n \geq N## ⇒ |(## a_n - l##|<ε . Show that if (## a_n ##) super converges to l then (## a_n...
  42. S

    Convergence and divergence of a series

    B]1. Homework Statement [/B] Find whether the series is convergent or divergent Homework Equations The Attempt at a Solution By ratio test I have, I would apply L'Hôpital's rule to find the value of limit but before that how do i simplify the expression? It has fractional part both in the...
  43. C

    Convergence of infinite sequences

    Homework Statement Let V consist of all infinite sequences {xn} of real numbers for which the series summation xn2 converges. If x = {xn} and y = {yn} are two elements of V, define (x,y) = summation (n=1 to infinity) xnyn. Prove that this series converges absolutely. Homework Equations The...
  44. E

    Series Convergence: Explaining P>1 & P>0

    Homework Statement Hi, everyone. I'd appreciate it if someone could explain something for me regarding the convergence of series. Thanks in advance![/B] Homework Equations In my calculus book, I'm given the following: (1) - For p > 1, the sum from n=1 to infinity of n^-p converges. (2) -...
  45. A

    MHB Improper Integral Question (convergence & evaluation)

    Hello, Two questions will be posed here. (1) Question about Convergence; quick way. Hello, I am trying to learn this concept on my own. My major question here is that, Is there a quick way, to tell if an integral converges or diverges? Suppose $\int_{0}^{\infty} \frac{x^3}{(x^2 +...
  46. B

    Pointwise convergence of Riemann integrable functions

    Hello Normally in order to change the order of limit and integration in rimann integration, you need uniform convergence. But let's say that you are not able to prove uniform convergence, but only pointwise convergence. And let's say you are able to prove that the functions are also...
  47. Julio1

    MHB What is the Radius of Convergence for the Power Series?

    Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$
  48. R

    MATLAB Solving Steady State Heat Conduction Eqn w/ MATLAB

    HI guys,this is my first programming experience , i have developed an MATLAB code for steady state heat conduction equation , on governing equation dt2 /dx2 + dt2/dy2 = -Q(x,y) i have solved this equation with finite difference method, As far as i know if we increase the mesh size it leads...
  49. C

    Definition of Convergence: Can n -> -infinity

    Homework Statement I've been given a question that makes use of 5^(n)*sin(pi*n!) The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity? So basically, I...
  50. I

    Convergence of iterative method and spectral radius

    Homework Statement Show that if given \mathbf{x}_0, and a matrix R with spectral radius \rho(R)\geq 1, there exist iterations of the form, \mathbf{x}_{n+1}=R\mathbf{x}_0+\mathbf{c} which do not converge. The Attempt at a Solution Let \mathbf{x}_0 be given, and let...
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