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.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Comparison test for convergence problem: why is this incorrect?

    Homework Statement The original question is posted on my online-assignment. It asks the following: Determine whether the following series converges or diverges: \sum^{\infty}_{n=1}\frac{3^{n}}{3+7^{n}} There are 3 entry fields for this question. One right next to the series above...
  2. C

    What is the Convergence Criterion for a Bounded Sequence with a Common Limit?

    Homework Statement Assume a_n is a bounded sequence with the property that every convergent sub sequence of a_n converges to the same limit a. Show that a_n must converge to a. The Attempt at a Solution Could I do a proof by contradiction. And assume that a_n does not converge...
  3. D

    Test for convergence of the series

    Homework Statement Q) Summation from 1 to infinity (1+(-1)^i) / (8i+2^i) This series apparently converges and I can't figure out why. Homework Equations The Attempt at a Solution (1+(-1)^i) / i(8+2^i/i) Taking the absolute value of the above generalization...
  4. J

    Proving the Minimum Radius of Convergence for a Sum of Taylor Series

    Homework Statement how to prove that radius of convergence of a sum of two series is greater or equal to the minimum of their individual radii i don't know how to begin, can someone give me some ideas?
  5. P

    MHB Testing Uniform Convergence of Complex Function Sequences with Natural Numbers

    How do I determine whether the following sequences of complex functions converge uniformly? i) z/n ii)1/nz iii)nz^2/(z+3in) where n is natural number
  6. L

    Convergence of a sequence of integrals

    Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...
  7. alexmahone

    MHB Analysis of Convergence for Series a) and b)

    Do the following series converge or diverge? a) $\displaystyle \sum\frac{1}{n!}\left(\frac{n}{e}\right)^n$ b) $\displaystyle\sum\frac{(-1)^n}{n!}\left(\frac{n}{e}\right)^n$ My attempt: a) $\displaystyle\frac{1}{n!}\left(\frac{n}{e}\right)^n\approx\frac{1}{\sqrt{2\pi n}}$ (Stirling’s formula)...
  8. C

    Does pointwise convergence imply convergence of integral on C[0,1]?

    Homework Statement Does pointwise convergence imply convergence of integral on C[0,1]? Homework Equations The Attempt at a Solution Would like to verify that f(t)=t^n is a counter example of this. Pointwise convergence goes to 0 on [0,1) and 1 on [1]. Norm(1) is unbounded. I...
  9. alexmahone

    MHB Is the Series $\displaystyle\sum\frac{\sin n}{n}$ Absolutely Convergent?

    Using the fact that the interval $\displaystyle\left[2k\pi+\frac{\pi}{4},\ 2k\pi+\frac{3\pi}{4}\right]$ contains an integer, prove that $\displaystyle\sum\frac{\sin n}{n}$ is not absolutely convergent.
  10. A

    Finding the Radius and Interval of Convergence of a Series

    Homework Statement Find the radius of convergence and interval of convergence of the series: as n=1 to infinity: (n(x-4)^n) / (n^3 + 1) Homework Equations convergence tests The Attempt at a Solution i tried the ratio test but i ended up getting x had to be less than 25/4 ...
  11. C

    Is the Sequence x_{n+1}=\sqrt{2x_n} Converging?

    Homework Statement Show that \sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}} converges and find the limit. The Attempt at a Solution I can write it also like this correct 2^{\frac{1}{2}},2^{\frac{1}{2}}2^{\frac{1}{4}},2^{\frac{1}{2}}2^{\frac{1}{4}}2^{\frac{1}{8}} so each time i...
  12. alexmahone

    MHB Absolute Convergence: Proving $\sum a_n$ is Convergent

    Prove that if $\displaystyle \left|\frac{a_{n+1}}{a_n}\right|\le\left|\frac{b_{n+1}}{b_n}\right|$ for $\displaystyle n\gg 1$, and $\displaystyle\sum b_n$ is absolutely convergent, then $\displaystyle\sum a_n$ is absolutely convergent.
  13. alexmahone

    MHB Convergence of $\displaystyle\sum\frac{n^5}{2^n}$

    Does the following series converge? $\displaystyle\sum\frac{n^5}{2^n}$
  14. alexmahone

    MHB When does the series $\sum_2^\infty\frac{1}{n(\ln n)^p}$ diverge for $p<0$?

    Test for convergence: $\sum_2^\infty\frac{1}{n(\ln n)^p}$ when $p<0$. My working: Consider the function $f(x)=\frac{1}{x(\ln x)^p}=\frac{(\ln x)^q}{x}$ when $q=-p>0$. In order to use the integral test, I have to establish that $f(x)$ is decreasing for $x\ge 2$. How do I proceed?
  15. alexmahone

    MHB Does the series $\sum_1\sin\left(\frac{1}{n}\right)$ converge?

    Test for convergence: $\sum_1\sin\left(\frac{1}{n}\right)$ My attempt: $\sin\left(\frac{1}{n}\right)\sim\frac{1}{n}$ Since $\sum_1\frac{1}{n}$ diverges, $\sum_1\sin\left(\frac{1}{n}\right)$ also diverges by the asymptotic convergence test. Is that correct?
  16. alexmahone

    MHB Subsequence - absolute convergence

    Let $\{a_n\}$ be a sequence, and $\{a_{n_i}\}$ be any subsequence. Prove that if $\sum_{n=0}^\infty a_n$ is absolutely convergent, then $\sum_{i=0}^\infty a_{n_i}$ is absolutely convergent. My attempt: $\sum |\ a_n|$ is convergent. $b_n=\left\{ \begin{array}{rcl}|a_{n_i}|\ &\text{for}& \...
  17. W

    Convergence in Galilean space-time

    To talk about differentiable vector fields in Galilean space-time, one needs to define convergence. Galilean space-time is an affine space and its associated vector space is a real 4-dimensional vector space which has a 3-dimensional subspace isomorphic to Euclidean vector space. There is no...
  18. alexmahone

    MHB Yes, that is correct.

    Prove: if $\sum a_n$ is absolutely convergent and $\{b_n\}$ is bounded, then $\sum a_nb_n$ is convergent. My working: $|b_n|\le B$ for some $B\ge 0$. $|a_n||b_n|<B|a_n|$ Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges. So, $\sum a_nb_n$ converges. (Absolute convergence...
  19. J

    Complex Power Series Radius of Convergence Proof

    Homework Statement If f(z) = \sum an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0. Homework Equations The Attempt at a Solution I know it is a power series and because R is positive I know it converges. And if...
  20. G

    Interesting convergence of sequence

    Homework Statement Let (a_n)_{n\in\mathbb{N}} be a real sequence such that a_0\in(0,1) and a_{n+1}=a_n-a_n^2 Does \lim_{n\rightarrow\infty}na_n exist? If yes, calculate it. Homework Equations The Attempt at a Solution I have a solution but I'd like to see other solutions..
  21. M

    Understanding L1 vs. Lp Weak Convergence

    Hello, I am struggling to understand why L1 is not weakly compact, while Lp, p>1, is. The example I have seen put forward is the function u_n (x) = n if x belongs to (0, 1/n), 0 otherwise, the function being defined on (0,1). It is shown this u_n converging to the Dirac measure, and...
  22. M

    Convergence Induction for Positive Sequences: Proving Limit Behavior

    Homework Statement If (an) — > 0 and \bn - b\ < an, then show that (bn) — > b Homework Equations The Attempt at a Solution
  23. C

    Does this sequence converge to the proposed limit?

    Homework Statement Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. a) lim \frac{1}{6n^2+1}=0 b) lim \frac{3n+1}{2n+5}=\frac{3}{2} c) lim \frac{2}{\sqrt{n+3}} = 0 The Attempt at a Solution A sequence a_n...
  24. N

    Order of Convergence: Iteration Scheme

    Homework Statement I have attached the actual problem as a picture (hope that is ok). Xn+1 = 12/(1+Xn), r=3 For each of the following iteration schemes the sequence xn will converge to the given value of r provided the initial value x0 is sufficiently close to r. In each case determine the...
  25. K

    Pointwise Convergence For Induced Distributions

    Homework Statement If U \subseteq \mathbb R^n find a sequence of locally integrable functions f_n \in L^1_{\text{loc}}(U) which converge pointwise, but whose induced distributions \langle f_n, \cdot \rangle: C_c^\infty(U) \to \mathbb R, \qquad \langle f_n, \phi \rangle = \int_U f_n \phi...
  26. M

    Test for convergence of a series

    Homework Statement Homework Equations The Attempt at a Solution I have no ideas how to continue. I also tried the comparison test but I don't know where to start. Please guide me...
  27. Also sprach Zarathustra

    MHB Uniform Convergence of $Y(x)$ in $(0,1]$

    Hello!A little problem:With the given series,$$Y(x)= \sum_{n=1}^{\infty}(-1)^n\frac{x^n\ln^nx}{n!} $$ ,why $Y(x)$ is Uniformly converges for all $x\in(0,1]$ ?Ok, I know that $Y(x)$ is u.c by M-test: $$\max{|x\ln{x}|}=\frac{1}{e}$$ And, $$ \sum_{n=0}^{\infty}\frac{(\frac{1}{e})^n}{n!} $$ Is...
  28. F

    Convergence of $\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n (z\in C)

    \sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n z in C. This only converges for z>\frac{-1}{2}, correct? Thanks.
  29. S

    Proving Convergence of {S_n/n} for Bounded Sequence {S_n}

    Homework Statement If {S_n} is a sequence whose values lie inside an interval [a,b], prove {S_n/n} is convergent. We don't know Cauchy sequence yet. All we know is the definition of a bounded sequence, and convergence and divergence of sequences. Along with comparison tests and Squeeze...
  30. C

    Complex Convergence with Usual Norm

    Homework Statement Determine whether the following sequence {xn} converges in ℂunder the usual norm. x_{n}=n(e^{\frac{2i\pi}{n}}-1) Homework Equations e^{i\pi}=cos(x)+isin(x) ε, \delta Definition of convergence The Attempt at a Solution I would like some verification that this...
  31. F

    Radius of Convergence for \sum_{n=2}^{\infty}z^n\log^2(n) in Complex Numbers

    \sum_{n=2}^{\infty}z^n\log^2(n), \ \text{where} \ z\in\mathbb{C} \sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2) By the ratio test, \lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right| \lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log...
  32. T

    I think this is a dominated convergence theorem question

    Hey All, I have the following integral expression: y = lim_{h\to0^{+}} \frac{1}{2\pi} \left\{\int^{\infty}_{-\infty} P(\omega)\left[e^{i\omega h} - 1 \right] \right\} \Bigg/ h And I am trying to understand when this expression will be zero. I was talking to a mathematician who said...
  33. estro

    Check Uniform Convergence of f_n(x): Solutions & Explanations

    Homework Statement f(x) is defined within [a,b]. f_n(x)=\frac{\big\lfloor nf(x) \big\rfloor}{n} Check if f_n(x) is uniform convergent. The Attempt at a Solution This one seems to be easy however since I didn't touch calculus for quite a time I'm not confident with my solution. |\frac...
  34. L

    Show Convergence of sequence (1 + c)(1 + c^2) (1 + c^n)

    Homework Statement Hey, so here is the problem: Suppose 0≤c<1 and let an = (1 + c)(1 + c2)...(1 + cn) for integer n≥1. Show that this sequence is convergent. Well I understand the basic concepts of proving convergence of sequences, but in class we've only ever done it with sequences where...
  35. C

    Convergence of {fn} wrt to C(X) metric

    Let C(X) be the set of continuous complex-valued bounded functions with domain X. Let {fn} be a sequence of functions on C. We say, that fn converges to f wrt to the metric of C(X) iff fn converges to f uniformly. By definition of the uniform convergence, for any ε>0 there exists integer N...
  36. A

    Trying to Prove Uniform Convergence: Analysis II

    Homework Statement I have a solution to the following problem. I feel it is somewhat questionable though If fn converges uniformly to f, i.e. fn\rightarrowf as n\rightarrow∞ and gn converges uniformly to g, i.e. gn\rightarrowf as n\rightarrow∞ , Prove that fngn...
  37. A

    Comparing Convergence: Exploring the p-Series Divergence for p<1

    Homework Statement \sum\frac{1}{n^{p}} converges for p>1 and diverges for p<1, p\geq0. The Attempt at a Solution (1) Diverges: I want to prove it diverges for 1-p and using the comparison test show it also diverges for p. \sum\frac{1}{n^{1-p}}=\sum\frac{1}{n^{1}n^{-p}}=\sum...
  38. C

    Show convergence using comparison test on sin(1/n)

    Homework Statement a) Test the following series for convergence using the comparison test : sin(1/n) Explain your conclusion. Homework Equations The Attempt at a Solution i must show f(x)<g(x) in order for it to converge other wise divergence. g(x) = 1/n sin(1/n) >...
  39. J

    Domain of convergence of Puiseux series

    May I ask if it's appropriate to ask in this sub-forum, are there any general methods for computing the domain of convergence for Puiseux series representations of algebraic varieties? I'm unable to find anything on the matter. For example, suppose I'm given the ideal in \mathbb{C}[z,w]...
  40. K

    Weak convergence of orthonormal sequences in Hilbert space

    So, I've found the result that orthonormal sequences in Hilbert spaces always converge weakly to zero. I've only found wikipedia's "small proof" of this statement, though I have found the statement itself in many places, textbooks and such. I've come to understand that this property follows...
  41. Fredrik

    Convergence in Measure: Understanding and Proving Almost Everywhere Convergence

    Not sure where to post about measure theory. None of the forums seems quite right. Suppose that ##(X,\Sigma,\mu)## is a measure space. A sequence ##\langle f_n\rangle_{n=1}^\infty## of almost everywhere real-valued measurable functions on X is said to converge in measure to a measurable...
  42. J

    Convergence of several improper integrals

    There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...
  43. B

    Convergence of C[0,2*pi] with f(x)=sin(x) and sup|f(x)|=1

    Folks, For C[0,2*pi] and given a function f(x)=sin(x) the supremum |f(x)|=max|f(x)| for x in [a,b] I calculate the sup|f(x)| to be = 0 but my notes say 1. The latter answer would be the case if f(x) was cos(x)...right?
  44. T

    Determining convergence or divergence

    Homework Statement Use a valid convergence test to see if the sum converges. Ʃ(n^(2))/(n^(2)+1) Homework Equations Well, according to p-series, I'd assume this sum diverges, but I don't know which test to use. The Attempt at a Solution I probably can't do this, but I was think...
  45. H

    How to increase the convergence speed?

    Hi, I'm using the finite difference method to solve both the Reynolds Equation and the Poisson equation in order to calculate the air pressure of a porous air bearing. The Reynolds equation describes the flow in the channel between the rotor and stator, while the Poisson equation the flow...
  46. T

    Divergence and convergence question

    Is the sum of 2 divergent series Ʃ(an±bn) divergent? From what I have learned is that it is not always divergent. Is this true? I believe that is what the picture i included is saying, but i maybe miss interpreting it. Also, is the product of 2 divergent series divergent or convergent?
  47. M

    Prove cauchy sequence and thus convergence

    Let (Xn) be a sequence satisfying |Xn+1-Xn| ≤ λ^n r Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.
  48. V

    Weak convergence of the sum of dependent variables, question

    Hi guys, Problem: Let {Xn},{Yn} - real-valued random variables. {Xn}-->{X} - weakly; {Yn}-->{Y} weakly. Assume that Xn and Yn - independent for all n and that X and Y - are independent. Fact that {Xn+Yn}-->{X+Y} weakly, can be shown using characteristic functions and Levy's theorem...
  49. T

    Difference between convergence of partial sum and series

    I am a little confused as to notation for convergence. I included a picture too. If you take a look it says "then the series Ʃan is divergent" Does the "Ʃan" just mean the convergence as to the sum of the series, or the lim an as n→ ∞ nth term? I believe it is the sum of the series but I...
  50. T

    Finding the radius of convergence

    Finding the radius of convergence... Homework Statement 1+2x+(4x^(2)/2!)+(8x^(3)/3!)+(16x^(4)/4!)+(32x^(5)/5!)+... Homework Equations I would use the ratio test. Which is... lim as n→∞ (An+1/An) The Attempt at a Solution I know what to do to find the answer, but I don't know...
Back
Top