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. P

    Convergence Properties: Exploring Series Convergence and Divergence

    Hello, I am trying to prove/disprove the following claims: (1) There exists a sequence a_n of positive number so that the series Ʃ a_n converges whereas the series Ʃ (a_n)^2 diverges. I believe I managed to disprove that. Is it indeed false? (2) There exists a sequence of real...
  2. P

    Convergence Test for Series with Cosine Cubed Terms

    Hi, How may I know whether the series ((-1)^n)[cos (3^n)x]^3/(3^n) converges/diverges?Should I use the Leibniz Criterion? It is stated that (cos a)^3 = (1/4)(3cos a + cos 3a)
  3. H

    Pick any test to determine convergence

    Homework Statement Use any method or test to see if the series converges or diverges. Homework Equations The series: ((1/n) - (1/n^2))^n The Attempt at a Solution Well the integral test won't work because there's no real integral for that according to Wolfram Alpha. Also if you...
  4. S

    Uniform convergence integration

    f is a continuous function on [0,infinity) such that 0<=f(x)<=Cx^(-1-p) where C,p >0 f_k(x) = kf(kx) I want to show that lim k->infinity ∫from 0 to 1 of f_k(x) dx exists so my idea is if I have that f_k(x) converges to f(x)=0 uniformly which I was able to show and that f_k(x) are all...
  5. B

    Does the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) converge?

    Homework Statement Determine whether the following series diverges, converges conditionally, or converges absolutely. \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) Homework Equations The Attempt at a Solution This was on today's test, and was the only problem I wasn't able to solve. I doubt my...
  6. S

    Convergence of Atomic Orbitals 2s and 2p with high Z number

    Hello all, this came up in my chemistry class when our prof. showed a graph of the size of atomic orbitals (or orbital energy) in relation to the Z number (or number of protons). He did this to show that as the z number increases the size of the orbitals also decrease (because effective nuclear...
  7. S

    Absolute convergence of series?

    The question is: Show that if \suman from n=1 to ∞ converges absolutely, then \suman2 from n=1 to converges absolutely. I'm not sure which approach to take with this. I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or...
  8. B

    Convergence of a sequence

    Homework Statement Does the following sequence converge, or diverge? a_{n} = sin(2πn) Homework Equations The Attempt at a Solution \lim_{n→∞} sin(2πn) does not exist, therefore the sequence should diverge? But it actually converges to 0? I appreciate all help thanks. BiP
  9. M

    Radius of convergence problem

    consider the rational function : f(x,z)=\frac{z}{x^{z}-1} x\in \mathbb{R}^{+} z\in \mathbb{C} We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for : \left | z\ln x \right |<2\pi Therefore, we consider an expansion around z=1 of the form...
  10. P

    Convergence implies maximum/minimum/both

    Hello, Could anyone please assist in proving that given a sequence converges it has a maximum/minimum/both? I have hitherto written that granted it converges it must be bounded and have a supremum and an infimum. Now, how may I proceed to prove that the latter are indeed within (the...
  11. D

    Convergence of the Sequence \sqrt[n]{n} to 1

    Homework Statement Be K \geq 1. Conclude out of the statement that \lim_{n \to \infty } \sqrt[n]{n} = 1, dass \sqrt[n]{K} = 1 The Attempt at a Solution \lim_{n \to \infty } \sqrt[n]{K} \Rightarrow 1 \leq \sqrt[n]{K} \geq 1 + ... I got issues with the right inequality...
  12. M

    Series Convergence and Sum Calculation

    Homework Statement Please write a specific function to define this series. Also provide a sum that the series converges to.Homework Equations Sn - {1, 1+1/e2, 1+1/e2+1/e4, 1+1/e2+1/e4+1/e6, ...} The Attempt at a Solution I know that the common ratio is 1/e2 and that you can raise that to...
  13. B

    Is the Proof Valid for the Convergence of the Sequence x_n?

    Homework Statement Let x_n be a convergent sequence with a ≤ x_n for every n, where a is any number. Prove that a ≤ lim x_n when n→∞. Homework Equations Definition of limit. The usual ε, N stuff. The Attempt at a Solution Let lim x_n = x and choose ε=x_n-a. Hence we have |x_n -...
  14. B

    Convergence of Finite Sets: A Limit on Repeated Elements?

    Homework Statement Let A be a finite subset of R. For each n in N, let x_n be in A. Show that if the sequence x_n is convergent then it must become a constant sequence after a while. Homework Equations The definition of limit. The Attempt at a Solution As A is finite, at least...
  15. M

    Use ratio test to find radius and interval of convergence of power series

    Homework Statement Use the ratio test to find the radius of convergence and the interval of convergence of the power series: [[Shown in attachment]] Homework Equations an+1/an=k Radius of convergence = 1/k Interval of convergence: | x-a |∠ R The Attempt at a Solution I...
  16. R

    Convergence of Integral: How to Prove for 0<k<1?

    Homework Statement Show that \int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx converges if 0<k<1 Homework Equations None The Attempt at a Solution Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one...
  17. E

    Investigating the convergence of a sequence

    Homework Statement Study the convergence of the following sequences a_{n} = \int^{1}_{0} \frac{x^{n}}{1+x^{2}} b_{n} = \int^{B}_{A} sin(nx)f(x) dx The Attempt at a Solution For the first one, I said it was convergent. I'm not exactly sure why though, my reasoning was...
  18. Square1

    Taylor Series Interval of COnvergence and Differention + Integration of it

    OK... "A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence" When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact? Does this come into play later in a...
  19. O

    MHB Can we use the fact that $L>1$ to show that the sequence is unbounded?

    Hello everyone! I am told that the limit of $\frac{x_{n+1}}{x_n}$ is $L>1$. I am asked to show that $\{x_n\}$ is not bounded and hence not convergent. This is what I got so far: Fix $\epsilon > 0$, $\exists n_0 \in N$ s.t. $\forall n > n_0$, we have $|\frac{x_{n+1}}{x_n}-L|<\epsilon$...
  20. P

    Series Test for Convergence Problem

    1. Is 1/(√(2n-1) convergent? 2. I have tried the first comparison test: an= 1/(√(2n-1) and bn=1/(n1/2. 0<=an<=bn. But bn diverges so we get no information. I have tried the second comparson test and let bn=1/n. But an/bn=∞ so once again I get no information. I have tried the ratio...
  21. G

    How fast does convergence have to be

    Reading through a real analysis textbook I noticed that \sum 1/K diverges but \sum 1/K1+\epsilon converges for all \epsilon > 0. This is confusing because 1/k will eventually be equally as small as the terms in 1/K1+\epsilon and therefore it should also converge. It may take much longer but...
  22. M

    Finding convergence of a recursive sequence

    Homework Statement x_{n+1} = (x_{n} + 2)/(x_{n}+3), x_{0}= 3/4Homework Equations The Attempt at a Solution I've worked out a few of the numbers and got 3/4, 11/15, 41/56, 153/209, ... It seems to be monotone and bounded below indicating it does converge I think. I need help figuring out what...
  23. D

    MHB Mean square convergence of Fourier series

    What is the statement of the mean square convergence of Fourier series?
  24. D

    MHB Prove Schwarz's & Triangle Ineqs for Inf Seqs: Abs Conv

    Prove the Schwarz's and the triangle inequalities for infinite sequences: If $$ \sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad \sum_{n = -\infty}^{\infty}|b_n|^2 < \infty $$ then $\sum\limits_{n = -\infty}^{\infty} a_nb_n$ converges absolutely. To show this, wouldn't I need to...
  25. M

    Don't understand convergence as n approaches infinity

    Here's the deal... I don't understand the limit as n→∞ of [(1+(.05/n))^20n -1]/[.05/n] My Calculus book says that it's supposed to approach {e^[(.05)(20)]-1}/[.05], but the numerator is a constant while the denominator goes to 0 as n→∞. The textbook, by Dr. Gilbert Strang, has similar limits...
  26. S

    Uniform Convergence: Intuitive Explanation

    I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: \int\sum b_{j}z^{j}dz=\sum\int b_{j}z^{j}dz=\int f(z)dz On an intuitive level I...
  27. S

    Prove Uniform Convergence of f_n=sin(z/n) to 0

    Homework Statement I need to show that f_{n}=sin(\frac{z}{n}) converges uniformly to 0. Homework Equations So I need to find K(\epsilon) such that \foralln \geq K |sin(\frac{z}{n})|<\epsilon I'm trying to prove this in an annulus: \alpha\leq |z| \leq\beta The Attempt at a Solution I'm having...
  28. U

    Interval of convergence.

    Homework Statement Find the radius and the interval of convergence for the series: Ʃ((-1)nxn)/(4nlnn) with sum n=2 to ∞. Homework Equations The Attempt at a Solution I'm testing for the left and right side of the interval. I've found that cn=1/(lnn), an=1/4, and R=1. I used...
  29. S

    Determining Convergence of a Sequence: Monotonicity & Boundness

    [b]1. Homework Statement [/b I want to see if a sequence converges by deciding on monotonicity and boundness. The sequence is: an=(n+1)/(2n+1) How to I go about determining if it converges or not based on those two factors? I am lost on how to go about it. THanks for any help...
  30. U

    Testing for series convergence.

    Homework Statement \sum(\frac{2n}{2n+1})n2 (The sum being from n=1 to ∞). Homework Equations The Attempt at a Solution Used exponent properties to get (\frac{2n}{2n+1})2n. Using the root test, the nth root of an = lim n->∞(\frac{2n}{2n+1})2 = 1. However, the root test is...
  31. N

    Convergence of the Expected Value of a Function

    Suppose that nx is binomially distributed: B((n-1)p, (n-1)p(1-p)) I wish to find the expected value of a function f(x), thus \sum_{nx=0}^{n-1} B() f(x) Assume that f() is non-linear, decreasing and continuous, f(x) = c is [0,1] to [0, ∞) I want to show that the above sum converges to f(p)...
  32. S

    Proving Convergence to 0

    Homework Statement If sequence an diverges to infinity and sequence an*bn converges then how do I prove that sequence bn must converge to zero? Homework Equations The Attempt at a Solution I really don't know how to go about this so any help would be so appreciated. Thanks
  33. J

    Convergence of indicator functions for L1 r.v.

    Hi all, I wonder if the following are equivalent. 1) E(|X|) < infinity 2) I(|X| > n) goes to 0 as n goes to infinity (I is the indicator function) 3) P(|X| > n) goes to 0 as n goes to infinity. 1) => 2) and 2) => 3) are easy to see, please help me to show 2) => 1) and 3) => 2)...
  34. S

    Convergence of an Improper Integral

    Let f(x) be a continuous functions on [0,∞) and that ∫ |f(t)|^2dt is convergent for 0≤t<∞. Let ∫ |f(t)|^2dt for 0≤t<∞ equals F. Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F. I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone...
  35. B

    Another question about Fourier series convergence

    I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...
  36. B

    Convergence of a factorial function

    The sum of "(n+3)!/(3n+2)!" with n=1 to n=inf. How do I find if it converges or diverges by using one of the tests(ratio, roots series, divergence, etc)?
  37. B

    Pointwise Convergence of Fourier Series for a continuous function

    Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...
  38. M

    Uniform Convergence of Power Series

    Given a power series \sum a_n x^n with radius of convergence R, it seems that the series converges uniformly on any compact set contained in the disc of radius R. This might be a silly question, but what's an example of a power series that doesn't actually also converge uniformly on the whole...
  39. M

    Advanced Calculus Sequence Convergence

    Homework Statement Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0. Homework Equations The Attempt at a Solution It's an if and only if proof, but I'm not sure how to prove it. Please help!
  40. B

    Is the Integral \int^{2}_{0}\frac{dx}{1-x^{2}} Convergent?

    Homework Statement Test the following integral for convergence \int^{2}_{0}\frac{dx}{1-x^{2}} Homework Equations The Attempt at a Solution So far I have brought it down to \int^{2}_{0}\frac{1}{1-x}+\frac{1}{1+x} dx However, it seems that this integral produces a...
  41. B

    Testing for integral convergence

    Homework Statement Test the following integral for convergence: \int^{∞}_{-∞}\frac{dx}{\sqrt{x^{4}+1}} Homework Equations The Attempt at a Solution I was able to use the ratio test to show that the integral converges if and only if \int^{∞}_{-∞}\frac{dx}{x^{2}} converges, but I haven't...
  42. K

    Is this proof for convergence of 3^n/n rigorous enough?

    Hi, I am trying to self study analysis and was practicing some problems. I wasn't sure if this solution to one of the problems I came across was rigorous enough. Basically, by writing down the first few terms of 3^n and n!, I figured I can say 3^n < 3*(n-1)! for all n>=13...without...
  43. S

    Does the convergence of {bn} to 0 guarantee the convergence of {anbn} to 0?

    Homework Statement Consider sequences {an} and {bn}, where sequence {bn} converges to 0. Is it true that sequence {anbn} converges to 0? The Attempt at a Solution Proof. First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2 {1, 2, 3, . . .}...
  44. B

    Convergence Proof for xn/xn+1: Need Help!

    Homework Statement If xn-> ∞ then xn/xn+1 converges. Homework Equations The Attempt at a Solution I can see why the statement is true intuitively, but do not know how to make a rigorous proof. I have looked at the definitions of divergence/convergence but can get any ideas of...
  45. B

    What does the Interval of Convergence for a power series tell me?

    So I know how to find the "Interval of Convergence" for a power series representation of a Function f(x). But I Still don't know what that "Interval of Convergence" does for me other than I can choose a number in it and plug it into the series. For Example e^{x}=\sum^{∞}_{n=0}...
  46. J

    Conditional vs unconditional convergence

    NOT talking about nonabsolute vs absolute convergence. I'm talking about conditional convergence. In my analysis text, this was a bit that was covered as enrichment and it straight up blew my mind. I don't get it. How can you simply rearrange terms and come up with a separate sum? They showed a...
  47. G

    Convergence of sequence : x + cosx

    Homework Statement xn+1 = xn + cosxn , n>=1 where x0 E [π/4 , 3π/4] = D. Show it converges, find rate of convergence.Homework Equations contraction theoremThe Attempt at a Solution Setting a function f(x) = x+cosx we have f'(x) = 1 - sinx, f''(x)= -cosx. Now f' >= 0, so f is increasing. For...
  48. K

    Interval of Convergence

    1. Find the radius and the interval of convergence for the series: Ʃ n=2 --> inf : [(-1)nxn]/ [4nln(n)] 2.To find the radius, we use the alternating series test. **an+1/an 3. From the alternating series test I find that the limit as n --> inf = 4. So our radius is 4...
  49. R

    Radius of convergence without complex numbers

    Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented. In examining the function f(x) = \frac{1}{1 + x^2} we can derive the series expansion \sum_{n=0}^\infty (-1)^n x^{2n} We...
  50. B

    Measurable and Unif. Convergence in (a,b)

    Hi, All: If {f_n}:ℝ→ℝ are measurable and f_n-->f pointwise, then convergence is a.e. uniform. Are there any conditions we can add to have f_n-->f in some open interval (a,b)? Correction: convergence happens in some subset of finite measure; otherwise above not true.
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