Convergence Definition and 1000 Threads

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  1. phosgene

    Convergence of \sum{\frac{2^n}{3^n - 1}} using the limit comparison test

    Homework Statement Use the limit comparison test to determine whether the following series converges or diverges. \sum{\frac{2^n}{3^n - 1}} Where the sum is from n = 1 to n = ∞. Homework Equations The limit comparison test: Suppose an>0 and bn>0 for all n. If the limit of an/bn=c, where...
  2. M

    Exploring the Convergence of \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)}

    Sequence Convergence \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)} I have tried some comparisons bot not conclude: \sum_{n=2}^\infty\frac{(-1)^n}{ln(n)}<=\sum_{n=2}^\infty\frac{(-1)^n}{1} \sum_{n=2}^\infty\frac{(-1)^n}{x-1}<=\sum_{n=2}^\infty\frac{(-1)^n}{ln(n)} Somebody having any insights...
  3. F

    Convergence of a sequence + parametre

    Homework Statement let ##a_n## be ##a_{n+1}=\frac{1}{4-3a_n} \quad n≥1## for which values of ##a_1## does the sequence converge? which is the limit? The Attempt at a Solution ##0<a_1<\frac{4}{3}## because if ##a_1>\frac{4}{3}→a_2<0## not possible. Now let's assume ##a_n## converges to M. I...
  4. C

    About interesting convergence of Riemann Zeta Function

    Hi, I was playing with Riemann zeta function on mathematica. I encountered with a quite interesting result. I iterated Riemann zeta function for zero. (e.g Zeta...[Zeta[Zeta[0]]]...] It converges into a specific number which is -0.295905. Also for any negative values of Zeta function, iteration...
  5. A

    Radius of convergence (power series) problem

    Homework Statement Ʃ (from n=1 to ∞) (4x-1)^2n / (n^2) Find the radius and interval of convergenceThe Attempt at a Solution I managed to do the ratio test and get to this point: | (4x-1)^2 |< 1 But now what? How do you get the radius and interval? Any help will be appreciated! Thanks
  6. F

    Convergence of a Sequence: Proving Existence of Limit Using Cauchy Sequences

    I think the solution I've found makes sense, but I'd like it to be double-checked. Homework Statement Let ##(a_n)## be a limited sequence and ##(b_n)## such that ##0≤b_n≤ \frac{1}{2} B_{n-1} ## Prove that if ##a_{n+1} \ge a_{n} -b_{n}## Then ##\lim_{n\to \infty}a_n## exists...
  7. H

    Radius and Interval of Convergence

    Homework Statement Find the radius and interval of convergence for the two series: 1) [(n+1)/n]^n * (x^n), series starting at n=1. 2) ln(n)(x^n), series starting at n=1. Homework Equations You're usually supposed to root or ratio your way through these. The Attempt at a...
  8. M

    Analytic Functions and Intervals of Convergence

    Working out of Boas' Mathematical Methods in the Physical Sciences; Chapter 14, section 2, problem 42... I'm supposed to write the power series of the following function, then find the disk of convergence for the series. Boas goes on to state, "What you are looking for is the point nearest...
  9. B

    Need Help finding Radius & Interval of convergence

    ƩHomework Statement Determine the radius of convergence and the interval of convergence for the follwing function expanded about the point a=2. f(x)= ln(3-x) Homework Equations ln(1-x) = Ʃ (x^n+1)/n+1 n=0 which has radius of convergence at |x|<1 The Attempt at a Solution...
  10. D

    MHB Convergence of Series with Logarithmic Terms

    $\sum\limits_{n = 2}^{\infty}\frac{1}{(\ln n)^{\ln n}}$ I am trying to show that this series diverges or converges
  11. J

    Estimating convergence of GRACE twin-satellites due to gravitational mass

    Homework Statement Hello noble physicists, I am struggling to solve a problem with any sort of confidence whatsoever, so to you I turn in the hopes of guidance. The problem refers to GRACE twin satellite convergence due to gravitational anomalies. I’d like to estimate the...
  12. 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...
  13. 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)
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. B

    Does the Sequence a_n = sin(2πn) Converge or Diverge?

    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
  20. M

    What Is the Radius of Convergence for This Rational Function Expansion?

    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...
  21. 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...
  22. 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...
  23. 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...
  24. 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 -...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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 $|\frac{x_{n+1}}{x_n}-L|<\epsilon$...
  31. 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...
  32. 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...
  33. 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...
  34. D

    MHB Mean square convergence of Fourier series

    What is the statement of the mean square convergence of Fourier series?
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. U

    What Is the Interval of Convergence for the Series Summation?

    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...
  40. 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...
  41. 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...
  42. 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)...
  43. S

    Does bn Converge to Zero if an Diverges and an*bn Converges?

    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
  44. 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)...
  45. 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...
  46. 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...
  47. 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)?
  48. 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...
  49. 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...
  50. 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!
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