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

    How Do You Determine the Interval of Convergence for a Series?

    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...
  2. 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...
  3. 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.
  4. B

    DE: Lower Bound for radius of convergence

    Prb:(x^4+4*x^2+16)y"+4(x-1)y'+6xy=0 P=(x^4+4*x^2+16) Q=4(x-1) R=6x P=0 for - 1 - 3^(1/2)*i 1 - 3^(1/2)*i - 1 + 3^(1/2)*i 1 + 3^(1/2)*i Q=0 for 1 R=0 for 0 Do we ignore Q & R, plotting P, then find shortest distance which would equal 2?
  5. H

    Uniform Convergence of Poisson Kernel on [-π, π] minus (-a, a)

    Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...
  6. W

    Monotone convergence - help required

    Hi all, http://www.scribd.com/doc/100079521/Document-1 Actually, I am trying to learn monotone convergence theorem, and I am stuck at one specific point, on the first page it says that ∫-∞→∞ f_n(x)dx = 1 for every n but the almost everywhere limit function is identically zero, what does it...
  7. M

    MHB Where am I going wrong with my Interval of Convergence calculations?

    Hello all, Again I find myself at odds with my online class. Somehow, and with two problems in a row, I am finding the reciprocal answer to what Math Lab is telling me. I would be very appreciative is someone could check my work. Find the limit of convergence, and the radius. \sum...
  8. H

    Linear control ODE - exponential convergence?

    Hello, I'm having hard times with the following simple linear ODE coming from a control problem: $$u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$$ with a given smooth α(t) satisfying $$0 \leq \alpha(t) \leq u(t) \quad\mbox{for all } t\geq 0.$$ My intuition is that $$\lim_{t\to\infty}...
  9. U

    Quotient Test for Convergence of Series

    Homework Statement The quotient test can be used to determine whether a series is converging or not. The full description is in the attachment. Homework Equations The Attempt at a Solution ( i ) Why must they both follow the same behaviour? Even if p ≠ 0, it says nothing about...
  10. T

    Alternating Series Test for Convergence

    Homework Statement Does this series converge absolutely or conditionally?Homework Equations Series from n=1 to ∞ (-1)^(n+1) * n!/2^n The Attempt at a Solution In trying to apply the alternating series test, I have found the following: 1.) n!/2^n > 0 for n>0 2.) Next, in testing to see if...
  11. M

    Evaluate Series Convergence Analytically

    Hi, while reading some artificial intelligence book, i came upon the following sum. How can I evaluate it analytically, so not guess it by computing many terms? It's easy to see by ratio test that it converges (intuitively too, since its a linear vs exponential function). \sum_{i=1}^\infty...
  12. B

    Uniform convergence and derivatives question

    In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}. What I don't like about the theorem is the huge amount of assumptions required: " Suppose that {fn} is a sequence of functions which are differentiable on...
  13. J

    Convergence of Unique Fourier Series

    Hi, I'm new to this forum, so I apologize if my LaTeX looks messed up. 1. Find the Fourier Series for f(x) = \sqrt{|x|} and prove it converges to f(x) 3. So, I've thus far proved that \sqrt{|x|} is piecewise continuous by proving that the limit as x approaches 0 (from both the right and left)...
  14. C

    Infinite series convergence question:

    Homework Statement Does \sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } converge \forall |\alpha |<e and if so, how can I prove it? Homework Equations { e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } The Attempt at a Solution...
  15. R

    Fourier Series Convergence at the Origin

    Homework Statement The Attempt at a Solution Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5
  16. R

    Understanding Fourier Series Convergence: Common Confusions Addressed

    Homework Statement This series is what dictates the graph above. The Attempt at a Solution I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi? I also don't get why terms...
  17. Jalo

    Solve Serie Convergence: Cauchy, d'Alembert & More

    Homework Statement Hi. I'm trying to solve a serie: Ʃ1∞2n+1*(n+1)! / (n+1)n+1 Homework Equations The Attempt at a Solution I tried solving it with Cauchy's method, but it failed. I also tried using d'alembert criterion, which game me the answer 2, so it should be divergent. However in the...
  18. A

    Jacobi method convergence for hpd matrices

    Homework Statement Let A be a squared, hermitian positive definite matrix. Let D denote the diagonal matrix composed of the diagonal elements of A, i.e. D = diag((A)11,(A)22,...(A)nn). Prove that if the Jacobi iterative method converges for A, then 2D - A must also be hermitian positive...
  19. Q

    Convergence in the sense of distributions

    I have the following problem: prove that the sequence e^{inx} tends to 0, in the sense of distributions, when n\to \infty. Here it is how I approached the problem. I have to prove this: \lim \int e^{inx}\phi(x)\,dx=0 , where \phi is a test-function. I changed variable: nx=x' and got...
  20. M

    Proof of Convergence by Integral Test and/or Comparison Test

    Note: This is not strictly a homework problem. I'm just doing these problems for review (college is out for the semester) - but I wasn't sure if putting them on the main part of the forum would be appropriate since they are clearly lower-level problems.(Newbie) Homework Statement The...
  21. D

    Series test for convergence or divergence

    I had a bit of trouble in testing series like this for convergence $$\sum_{ n=1 }^{ \infty } \frac { 1 }{ 2n+1 } $$ If by the comparison test, ##\frac{ 1 }{ 2n+1 } < \frac{ 1 }{ 2n }## for all of n>0, and ##\lim_{ n \rightarrow \infty} \frac{ 1 }{ 2n }## =0, then the series should be...
  22. S

    Proving Convergence of the Series $\sum_{r=1}^{\infty} \frac{r!}{3^{r^{2}}}$

    \sum_{r=1}^{\infty} \frac{r!}{3^{r^{2}}} My solution: \frac{3^{r^2}}{r!} > r^2 So \frac{r!}{3^{r^2}} < \frac{1}{r^2} So as 1/r^2 converges, it converges by comparison test. This was in my exam today, I messed up a lot leading up to it. But the question said I could use any test in...
  23. DryRun

    How Do You Determine the Convergence of Infinite Series?

    Homework Statement (a)\;\sum^{\infty}_{n=1}\frac{n-5}{n^2}\;(solved) (b)\;\sum^{\infty}_{r=1}\frac{2r}{1+r^2}\;(solved) (c)\;\sum^{\infty}_{n=1}\frac{\cos^4 nx}{n^2}\;(solved) (d)\;\sum^{\infty}_{n=1}\frac{3^r+4^r}{4^r+5^r} (e)\;\sum^{\infty}_{r=1}\frac{r^r}{r!}\;(solved)...
  24. T

    How to test this serie for convergence?

    Homework Statement I'm trying to determine if Ʃ 1/(3^ln(n)) converges. Homework Equations The Attempt at a Solution The preliminary test isn't of any help since lim n→∞ an = 0. I tried the integral test but I couldn't integrate the function, and I don't think it's the best...
  25. P

    Complex Analysis - Radius of convergence of a Taylor series

    Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...
  26. S

    Why does P-a-s Convergence Equal Limm→∞ P({Supn≥m|Xn-X| ≥ε })?

    So I have a definition; Xn n=1,2... is a sequence of random variables on ( Ω,F,P) a probability space, and let X be another random variable. We say Xn converges to X almost surely (P-a-s) iff P({limn →∞ Xn=X}C) = 0 It then goes on to say that checking this is the same as checking limm...
  27. C

    Series, find Divergence or Convergence

    Homework Statement Find the Divergence or Convergence of the series \sum^{∞}_{n=1}\frac{2n^2+3n}{\sqrt{5+n^5}} Homework Equations Ratio Test, Comparison Test, Limit Comparison Test, Integral test etc. The Attempt at a Solution This question was on my final exam and the only question of...
  28. P

    Comples analysis - Radius of convergence of a Taylor series question

    Homework Statement Find the radius of convergence of the Taylor series at z = 1 of the function: \frac{1}{e^{z}-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far. Multiply top and bottom by minus 1 to get: -1/(1-e^z) And then...
  29. B

    Solving ODE Convergence Problem with Secant Approximation

    I have a pesky problem, I have this function of time, S(t) and I'm trying to find how far to evaluate S (its an expensive process and must be done for finite t=time). Essentially, I want to measure S until dS/dt ≈ 0. But my current criteria is making the computation itself inefficient not to...
  30. K

    Absolute Convergence Theorem and Test for Divergence Connection

    Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity. Homework Equations Theorem for absolute convergence. Test for divergence The Attempt...
  31. K

    Determine Absolute Convergence, Conditionally Convergent, or Divergent

    Homework Statement ##\sum _{n=1}^{\infty }\dfrac {\left( -3\right) ^{n}} {n^{3}}## According to Wolfram Alpha the series diverges by the Limit Comparison Test, but I remember that the limit comparison only works with series greater than zero. How is this possible? Homework Equations...
  32. B

    Convergence or divergence (series)

    Homework Statement Ʃ[(-1)^n (cosn)^2]/√n The Attempt at a Solution i don't have the slightest clue where to start
  33. T

    Convergence of a series with tests

    Determine whether the series converges or diverges: \sum ln k/ k3 now I said that this series converges by the comparison test, using ln k / k since I know that goes to 0 Would that be the right logic?
  34. K

    Test the Series for Convergence or Divergence

    Homework Statement ##\sum _{n=1}^{\infty }\left[ \left( -1\right) ^{n}\right] \dfrac {\sqrt {n}} {1+2\sqrt {n}}##Homework Equations Alternating Series test, Absolute convergence theorem, p-series, and test for divergence. The Attempt at a Solution The alternating series test tells us that the...
  35. G

    Probability - almost sure convergence

    Homework Statement We have ##\mathbb{P}(X_n = 1) = p_n ## and ##P(X_n=0) = 1-p_n ## the question is about almost sure convergence. i.e. does ## X_n \overset{a.s.}{\longrightarrow} 0 ## if ##p_n = 1/n##? Homework Equations ##X_n \overset{a.s.}{\longrightarrow } X ## if ## \mathbb{P}(...
  36. G

    A limit in probability (possibly dominated or monotone convergence theorem)

    Homework Statement ## \sqrt{2\pi} a \exp({a^2 \over 2} ) P( \xi \geq a) \to 1 ## ## \xi \text{ ~ } N(0,1) ## Homework EquationsThis implies ##\sqrt{2\pi} a \exp({a^2 \over 2} ) \int_a^\infty a \exp({a^2\over 2}) \exp (-{x^2 \over2}) dx \to 1 ## The Attempt at a Solution The integral...
  37. K

    Cauchy sequnce and convergence of a non-monotonic sequence.

    Homework Statement Hello, I have a question concerning convergence of the non-monotonic sequences which takes place when the Cauchy criterion is satisfied. I understand that |a_n - a_m| <ε for all n,mN\ni Homework Equations What I don't see is how (a_{n+1} - a_n) →0is not...
  38. A

    How do you determine convergence of a series?

    I just need to know how you determine if a series of convergent or divergent. I have this example in which I know is divergent I just don't know why: summation (n=1 to infinity) 1/(2n) The first couple of terms are 1/2 + 1/4 + 1/6 + 1/8 + ... Up until that point, it's already beyond...
  39. DryRun

    Test series for convergence or divergence

    Homework Statement There are 3 parts to this problem: (a) \; \sum^{\infty}_{n=1} \frac{n^4}{4^n} (b) \; \sum^{\infty}_{n=1} \left( \frac{n+8}{n} \right)^n (c) \; \sum^{\infty}_{n=1} \frac{5^n-8}{4^n+11} The attempt at a solution (a) I've used the Ratio test. So, u_n=\frac{n^4}{4^n} and...
  40. K

    Determine Series' Convergence or Divergence

    Homework Statement ##\sum _{n=1}ne^{-n}## Homework Equations Ratio Test Integral Test The Attempt at a Solution I know that by the ratio test, it converges absolutely. But, I am unable to determine its convergence through the integral test . Could someone help? I thought that the...
  41. F

    Absolute Convergence of a Series

    Homework Statement I'm given this series and asked whether it converges, absolutely converges, or diverges. Ʃ(n=0 to infinity) [2(-1^n)(3^(n+1))]/5^n Homework Equations The Attempt at a Solution The answer states that it is absolutely convergent, and that it converges to 15/4. Everything...
  42. K

    Determine Convergence or Divergence. If conv. find the sum:

    Homework Statement ##\sum \dfrac {1+2^{n}} {3^{n}}## According to Wolfram Alpha the sum is 5/2. But, I think that my method is fine and shows another result. The Attempt at a Solution ##\sum \dfrac {1+2^{n}} {3^{n}}=\sum \left[ \left( \dfrac {1} {3}\right) ^{n}+\left( \dfrac {2} {3}\right)...
  43. J

    Convergence for a series: what is wrong with my method?

    Convergence for a series: what is wrong with my method?? Homework Statement For the following series, write formulas for the sequence an, Sn, and Rn, and find the limits of the sequences as n→∞. Homework Equations Sn is the partial sum of the series. Rn is the remainder and is...
  44. V

    Pointwise/Uniform Convergence proof

    Homework Statement Wondering if you guys could check my proof. This is my first problem with sequences of functions Let a > 0 and f_{n}(x) = \frac{nx}{1+nx}. a) Show that the sequence of functions (f_{n}) converges pointwise on [0,∞) b) Show that (f_{n}) converges uniformly on [a,∞)...
  45. T

    I with finding the radius of convergence of a complex series

    Radius of convergence of the series n^2(x^n)/(3n!) I am stumped the question is: find the radius and interval of convergence of the following series {sum_(n=1)^(Infinity)}((n^2)(x^n))/(3*6*9***3n) I'm assuming that equal to ((n^2)(x^n))/(3n)! then lim_(n->infinity) of...
  46. L

    Convergence of a Recursive Sequence

    Homework Statement A sequence {an} defined recursively by a1=1 and an+1=\frac{1}{2+a subn}, n\geq1. Show that the sequence is convergent. Homework Equations If a sequence is bdd below and decreasing or it is bdd above and increasing, then it is convergent. The Attempt at a Solution...
  47. B

    MHB Convergence of an Infinite series and a related Qn

    Dear friends, \sum_{x=1}^{\infty}\frac{1}{x} diverges. But \sum_{x=1}^{\infty}\frac{1}{x^{2}}=\frac{\pi^{2}}{6} How can we prove that \sum_{x=1}^{\infty}\left(\frac{1}{x^{\left(1+epsilon\right)}}\right) converges to a finite value? Thanks in advance. Bincy.
  48. F

    Finding the radius of convergence of a series.

    Homework Statement What is the radius of convergence of the Taylor Series of the function f(z) = z cot(z), at the point z = 0? Homework Equations Taylor series is given by: \sum_{k=0}^{\infty} \frac{f^{(k)}(z_{0})}{k!} (z - z_{0}) And the radius R by: \lim_{n \to \infty}...
  49. T

    Convergence of indeterminate forms of a sequence

    State whether the sequence converges as n--> ##∞##, if it does find the limit i'm having trouble with these two: n!/2n and ∫ e-x2 dx now I know they're special forms so the ordinary tricks won't work. Any help or hints?
  50. B

    Limit proof on Sequence Convergence

    Consider a sequence \{ a_{n} \} . If \lim_{n→∞}a_{n} = L Prove that \lim_{n→∞}a_{n-1} = L I am trying to use the Cauchy definition of a limit, but don't know where to begin. Thanks. BiP
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