What is Radius of convergence: Definition and 140 Discussions

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or



∞


{\displaystyle \infty }
. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

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

    Radius of Convergence for Series with (-1)^(n-1) and x^n Terms

    Homework Statement Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint. Homework Equations The Attempt at a Solution Not really sure how to start this. I know I'm supposed to use the...
  2. E

    Lower bound for radius of convergence of series solutions about a given point

    Homework Statement Determine a lower bound for the radius of convergence of series solutions about a) x_{0}=0 and b) x_{0}=2 for \left(1+x^{3}\right)y''+4xy'+y=0. Homework Equations N/A The Attempt at a Solution The zero of P\left(x\right)=\left(1+x^{2}\right) is -1. The...
  3. B

    Finding Radius of Convergence of the Power Series

    Homework Statement "Find the radius of convergence of the power series for the following functions, expanded about the indicated point. 1 / (z - 1), about z = i. Homework Equations 1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... + Ratio Test: limsup sqrt(an)^k)^1/k The...
  4. S

    Centre and Radius of Convergence

    Homework Statement Find the centre and radius of convergence: \stackrel{\infty}{n=1}\sum n.(z+i\sqrt{2})^{n} Homework Equations 1) Ratio test \left|\frac{a_{n+1}}{a_{n}}\right|<1 2) Textbook uses \stackrel{lim}{n-> \infty}\left|\frac{a_{n}}{a_{n+1}}\right| The Attempt at a Solution using...
  5. P

    Radius of convergence of complex power series

    Homework Statement Find the radius of convergence of the series: ∞ ∑ n^-1.z^n n=1 Use the following lemma: ∞ ∞ If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also...
  6. L

    Radius of Convergence for Moderately Complicated Series

    1. The problem statement: Show that the following series has a radius of convergence equal to exp\left(-p\right) Homework Equations For p real: \Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n} The Attempt at a Solution...
  7. F

    Power series, radius of convergence and Abel's Theorem

    Homework Statement Suppose the series \sum_{n=0}^{\infty} a_n x^n has radius of convergence R and converges at x = R. Prove that \lim_{x \to R^{-}}\large( \sum_{n = 0}^{\infty} a_n x^n \large) = \sum_{n = 0}^{\infty} \large( \lim_{x \to R^{-}} a_n x^n \large) 2. Question For the case R...
  8. S

    Radius of convergence quick help

    quick help on this i seem to be missing some logic or process determine the lower bound the radius of convergence of series solutions about the given X0 (2+x^2)y''-xy'+4y=0 xo=0
  9. A

    Taylor series radius of convergence and center

    When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R]. Does this imply that: 1. There also exists a Taylor series expansion...
  10. G

    Finding Radius of Convergence: Calculating (1/n!)x^(n!)

    Hey , I was wondering if anyone could help me out with this question regarding calculating the radius of convergence of the infinity series of (1/n!)x^(n!) This is my work First we consider when abs(x) < 1 then we have 0 <= abs(x^n!) <= abs(x^n) so we know that the series converges...
  11. Y

    Radius of convergence of the series

    Homework Statement The coefficients of the power series the sum from n=0 to infinity of an (x-2)^n satisfy ao=5 and an= [(2n+1)/(3n-1)] an-1 for all n is greater than or equal to 1. The radius of convergence of the series is A) 0 B) 2/3 C) 3/2 D) 2 E) infinite Homework Equations...
  12. I

    Understanding the Radius of Convergence for E 1/n^x in Calculus

    Having a hard time with this one: E 1/n^x , have tried too use n^-x=e^(-x ln n) which in turn e^(...) = lim n->OO (1-(x ln n)/n)^n and then go on with finding the centre, but I feel I'm far far off. How to get it like E an(x-c)^n and use the more straight foreward path.
  13. B

    Root vs. ratio in determining radius of convergence of a power series

    Hi everyone :smile: When determining the radius of convergence of a power series, when should I use the ratio (a[sub n+1] / a[sub n]) test versus the root (|a[sub n]|^(1/n)) test? I know that I'm supposed to use the ratio only when there are factorials, but other than that, are these tests...
  14. M

    Finding the Radius of Convergence for a Series with Exponential Growth

    Hi there - I'm trying to work out the radius of convergence of the series \sum_{n \geq 1} n^{\sqrt{n}}z^n and I'm not really sure where to get going - I've tried using the ratio test and got (not very far) with lim_{n \to \infty} | \frac{n^{\sqrt{n}}}{(n+1)^{\sqrt{n+1}}}|, and with the root...
  15. B

    Taylor series of real function with zero radius of convergence

    Can anyone please give me an example of a real function that is indefinitely derivable at some point x=a, and whose Taylor series centered around that point only converges at that point? I've searched and searched but I can't come up with an example:P Thank you:)
  16. J

    Radius of convergence question

    Homework Statement What feature of the ODE explains your value for the radius of convergence of the series y2? y2 is a series which satisfies the ODE and I found that it converges for \abs{2x^2} < 1. Homework Equations y2=x-\frac{2}{3}x^2-\frac{4}{15}x^5+ \cdots ODE...
  17. J

    Find the radius of convergence

    Homework Statement Find the radius of convergence of the Series: \sum_{i=1}^{\infty}\frac{(2n)!x^n}{(n!)^2} The attempt at a solution I used the Ratio Test but I always get L = |\frac{2x}{n+1}| The answer is 1/4. I think I am mistaking with factorial.
  18. P

    Understanding Radius of Convergence in Power Series: A Graphical Approach

    Hi. Not really a homework question. Just a doubt i would like to confirm. Is the radius of convergence of a power series always equal to the radius of convergence of it's primitive or when its differentiated? I have done a few examples and have noticed this. I am trying to understand this...
  19. B

    Calculating the Radius of Convergence for a Series Using the Ratio Test

    Homework Statement Find the radius of convergence for \Sigma \frac{nx^{2n}}{2^{n}} Homework Equations Ratio test The Attempt at a Solution I apply the ratio test to get \frac{(n+1)(x^{2})}{2n}. I let n approach infinity, to get \frac{1}{2}. So, this series converges when |x2|<1...
  20. R

    Finding Radii of Convergence Using Ratio Test

    Hi please could you assist me: questions posted below:Assuming the function f is holomorphic in the disk \[D(0,1) = \{ z \in \mathbb{C}:|z| < 1\}\], prove that \[g(z) = \overline {f(\overline z )} \] is also holomorphic in D(0,1) and find its derivative? Find the radii of convergence of the...
  21. E

    Region of Convergence for Series in x+2y: Description and Solution

    [SOLVED] radius of convergence Homework Statement Let D be th region in the xy plane in which the series \sum_{k=1}^{\infty}\frac{(x+2y)^k}{k} converges. Describe D.Homework Equations The Attempt at a Solution By the ratio test, we find the radius of converge of the series in x+ 2y to be 1...
  22. S

    Real Analysis - Radius of Convergence

    Homework Statement Suppose that \sumanxn has finite radius of convergence R and that an >= 0 for all n. Show that if the series converges at R, then it also converges at -R. Homework Equations The Attempt at a Solution Since the series converges at R, then I know that \sumanRn = M...
  23. rootX

    Solving for Radius of Convergence: 1/(1+x^2) and arctan (x)

    [SOLVED] Radius of Convergence Homework Statement 1/(1+x^2) = sum ( (-1)^k*x^(2k) ; 0 ; inf) - A integrating arctan (x) = sum ((-1)^k * x^(2k+1) / (2k+1) ; 0; inf) B I know A has radius of converge of 1, and I calculated B to be 2. My assignment solution says "Similarly, the...
  24. B

    What is the radius of convergence for this power series?

    I am looking for radius of convergence of this power series: \sum^{\infty}_{n=1}a_{n}x^{n}, where a_{n} is given below. a_{n} = (n!)^2/(2n)! I am looking for the lim sup of |a_n| and i am having trouble simplifying it. I know the radius of convergence is suppose to be 4, so the lim sup...
  25. A

    Radius of convergence for (1+x)^1/2

    Problem Statement: Compute the Taylor Series for (1+x)^1/2 and find the radius of convergence Problem Solution: The Taylor Series expansion I get is T(x) = 1 + (0.5*x) - (0.25*x^2)/2 + (0.375*x^3)/3! - (0.9375*x^4)/4! +...-... So to get radius of convergence I have to find a...
  26. H

    Positive Radius of Convergence

    Problem: Suppose that {a_{k}}^{\infty}_{k=0} is a bounded sequence of real numbers. Show that \suma_{k}x^{k} has a positive radius of convergence. Work: I have attempted to use the ratio test and failed. I am suspicious I can try the root test, but I am not sure how to work it...
  27. M

    Radius of Convergence for (-1)^n(i^n)(n^2)(Z^n)/3^n in Complex Analysis

    Homework Statement Find the radius of convergence of (-1)^n(i^n)(n^2)(Z^n)/3^nThe Attempt at a Solution i have got to lZl i (n+1)^2/3n^2 but am unsure how to complete it...
  28. J

    Radius of convergence of an infinite summation

    [SOLVED] radius of convergence of an infinite summation Homework Statement find the radius of convergence of the series: \sum\frac{(-1)^k}{k^2 3^k}(z-\frac{1}{2})^{2k} Homework Equations the radius of convergence of a power series is given by \rho=\frac{1}{limsup |c_k|^{1/k}}...
  29. K

    What Are the Radii and Intervals of Convergence for These Series?

    Radius of convergence, interval of convergence Homework Statement Find the radius of convergence and the interval of convergence of the following series. a) \sum_{n=0}^\infty \frac{x^n}{(n^2)+1} c) \sum_{n=2}^\infty \frac{x^n}{ln(n)} e) \sum_{n=1}^\infty \frac{n!x^n}{n^2} f)...
  30. J

    Prove about radius of convergence

    Homework Statement Prove that the radius of convergence \rho of the power series \sumck (z-a)^k over all k, equals 1/R when ck is not 0 and you know that: |\frac{ck+1}{ck}|=R>0 Homework Equations I was planning on using that the radius of convergence is in this case: \rho=...
  31. K

    Radius of Convergence: f(x) = x^4 / (2 - x^4)

    Homework Statement f(x) = x^4 / (2 - x^4). Specify radius of convergence. Homework Equations Power Series f`(x) = c2 + 2c2(x-a) + 3c3(x-a)^2 + ... = (infinity)sigma(n=1) [n * cn * (x-a)^(n-1)] The Attempt at a Solution I'm not sure what to do. Usually, most problems are like x^3 /...
  32. C

    Converging Power Series: Finding the Radius of Convergence for (3x+4)^n/n

    [b]1. The radius of convergence of the power series the sum n=1 to infinity of (3x+4)^n / n is a 0 b 1/3 c 2/3 d 3/4 e 4/3 [b]2. the sum n=1 to infinity of (3x+4)^n / n [b]3. no idea do the ration test to get abs value 3x+4 < 1 ?
  33. B

    Determining the Radius of Convergence for a Series with Limited Limit Points

    Could someone please help me out with the following? I need to determine the radius of convergence of the following series. It is exactly as given in the question. \sum\limits_{n = 0}^\infty {\left( {3 + \left( { - 1} \right)^n } \right)^n } z^n The suggestion is to use the...
  34. B

    Radius of convergence of power series

    Homework Statement Find the radius of convergence of the following series. \sum\limits_{k = 1}^\infty {2^k z^{k!} } Homework Equations The answer is given as R = 1 and the suggested method is to use the Cauchy-Hadamard criterion; R = \frac{1}{L},L = \lim \sup \left\{ {\left|...
  35. J

    Is it possible for radius of convergence to be negative?

    is it possible for "R" (radius of convergence) to be negative? is it possible for "R" (radius of convergence) to be negative? for example: -|x|<1 and R=-1?
  36. C

    Power series and finding radius of convergence

    Homework Statement "Find the radius of convergence and interval of convergence of the series" \sum_{n=0}^\infty \frac{x^n}{n!} Homework Equations Ratio Test The Attempt at a Solution \lim{\substack{n\rightarrow \infty}} |x/n+1| (I can't seem to get the |x/n+1| to move up where it should be)...
  37. B

    Maclaurin Series used to find associated radius of convergence Q

    I have the Maclaurin series for cos (x), is their a way to find its radius of convergence from that? ALSO Is there a trick to find the shorter version of the power series for the Maclaurin series, I can never seem to find it so instead of the long series with each term but like E summation (the...
  38. B

    Analyzing a Power Series: Convergence and Radius of Convergence

    I really need help with this exercise. Consider the power series \sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}. for z\in\mathbb{C}. I need to answer the following questions: a) Is the series convergent for z = 1? This is easy; just plug in z = 1 and observe that the alternating...
  39. M

    Does the exponent in a series affect its radius of convergence?

    Hi folks. I need to find the radius of convergence of this series: \sum_{k=0}^\infty \frac{(n!)^3z^{3n}}{(3n)!} The thing throwing me off is the z^{3n}. If the series was \sum_{k=0}^\infty \frac{(n!)^3z^n}{(3n)!} I can show it has radius of convergence of zero. But z^{3n} means its only...
  40. B

    Exploring the Radius of Convergence of a Complex Series

    I am given this series: \sum_{n=1}^\infty\frac{2n}{n^2+1}z^n. First I have to find the radius of convergence; I find R = 1. Then I have to show that the series is convergent, but not absolutely convergent, for z = -1, i.e. that the series \sum_{n=1}^\infty(-1)^n\frac{2n}{n^2+1} is...
Back
Top