Radius of convergence Definition and 140 Threads

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.

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  1. 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...
  2. 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?
  3. 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...
  4. 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...
  5. 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...
  6. 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}...
  7. M

    Help me find the radius of convergence?

    Homework Statement Ʃn!(x-1)n I need to find the radius of convergence for this summation from n=0 to n=∞ The Attempt at a Solution I started off with the ratio test: (n!(n+1)(x-1)(x-1)n)/(n!(x-1)n) = (n+1)(x-1) (x-1)lim(n+1)...Now at this point it looks to me like the series does...
  8. L

    Radius of Convergence of power series anx^n^2

    Homework Statement Suppose that the power series \sumanxn for n=0 to n=∞ has a radius of convergence R\in(0,∞). Find the radii of convergence of the series \sumanxn2 from n=0 to n=∞ and \sumanx2n.Homework Equations Radius of convergence theorem: R = 1/limsup|an|1/n is the radius of...
  9. T

    Radius of convergence question

    Homework Statement Suppose that the following series converges when x = -4 and diverges when x = 6. ∑{n=0 -> ∞} c_n • x^n What is the interval of convergence? The Attempt at a Solution I think it is [-5,5) but my friend reckons that it is [-5,6). I don't think [-5,6) is correct because this...
  10. S

    Trying to find the radius of convergence of this complicated infinite series

    Homework Statement k is a positive integer. \sum^{\infty}_{n=0} \frac{(n!)^{k+2}*x^{n}}{((k+2)n)!} Homework Equations The Attempt at a Solution I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get...
  11. J

    Extending radius of convergence by analytic continuation

    Hi, Suppose I have an analytic function f(z)=\sum_{n=0}^{\infty} a_n z^n the series of which I know converges in at least |z|<R_1, and I have another function g(z) which is analytically continuous with f(z) in |z|<R_2 with R_2>R_1 and the nearest singular point of g(z) is on the circle...
  12. 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?
  13. 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...
  14. 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...
  15. 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...
  16. T

    Using ratio test to find radius of convergence

    Homework Statement Ʃ((x-3)^(n)) / (n*2^(n)) Homework Equations lim as n→ ∞ (An+1 / An) The Attempt at a Solution When dividing two fractions, invert the second and multiple to get what you see below. (x-3)^(n+1)/((n+1)*2^(n+1)) * (n*2^(n))/((x-3)^(n)) Do some cross...
  17. J

    Question about radius of convergence of fractional power series

    Suppose I have the Laurent series with region of convergence given below: f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5} Can I conclude the Laurent-Puiseux series: f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n has a region of convergence...
  18. P

    Radius of convergence: 1/(1+x^2) about 1, using only real analysis

    I've seen this thread: https://www.physicsforums.com/showthread.php?t=297842 and that is the exact question I need to to answer. What is the radius of convergence of 1/(1+x^2) expanded about x_0=1? The problem is, I can only use an argument in real analysis. I see the answer is...
  19. P

    Tricky series radius of convergence question (analysis course)

    Homework Statement Find the radius of convergence of sum from 1 to n of 1/(n^n) * x^(2^n) Homework Equations The Attempt at a Solution Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent
  20. nomadreid

    Radius of convergence around a=2 for ln(x)

    When I use the calculation from Wikipedia that says that the radius of convergence of a series is lim as n goes to infinity of |an/an+1|, I get for the Taylor series expansion of ln(x) around a=2 the answer of an infinite radius of convergence, which would mean that it would be valid everywhere...
  21. I

    Radius of convergence (Power series)

    Homework Statement Hi there, I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.Take for example: The...
  22. N

    Complex analysis, taylor series, radius of convergence

    Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. Homework...
  23. G

    Question on Radius of Convergence for values of x, when f(x) is x^2

    Homework Statement This is not so much an entire problem I need help with but just a part. It is a power series where after you do the ratio test, you end up with |4x^(2)| < 1, so |x^(2)| < 1/4. Since the radius of convergence is |x-a| < R, I end up with -1/4 < x^(2) < 1/4, but because...
  24. M

    Find the Summation Notation and the Radius of Convergence

    Homework Statement Find the Summation Notation and Radius of Convergence of this series. 5, x, 10, x, ... The Attempt at a Solution I don't know how did they come up with that equation.. But the summation seems right.. Can anyone tell me how did they arrive with that equation? I've tried...
  25. S

    Finding radius of convergence of series ?

    Homework Statement How would I find the radius of convergence of this series? f(x)=10/(1-3x)2 is represented as a power series f(x)=\sum from n=0 to \infty CnXn Homework Equations The Attempt at a Solution Okay so I tried deriving, using d/dx(1/1-3x)=3/(1-3x)2 and ended up with...
  26. S

    Finding the radius of convergence and interval of convergence

    Homework Statement This is the question of mine that I'm having a little confusion about. I know the whole process in which you use the ratio test to determine the radius of convergence and using that you test the end points of the summation to see if they converge at the end points aswell...
  27. K

    Help Convergence of Power Series, interval and radius of convergence question

    Homework Statement Determine the radius of convergence, the interval of convergence, and the sum of the series Summation from k=2 to ∞ of k(x-2)^k+1. Homework Equations ratio test? The Attempt at a Solution possibly take the derrivitive of the power series, then find the sum then integrate...
  28. S

    Power Series: Find Interval & Radius of Convergence

    Homework Statement \Sigma (from index k = 1 until infinity) Within the Sigma is the series : (k! * (x^k)) Homework Equations Ratio Test : lim as k approaches infinity |a(k+1) / ak| The Attempt at a Solution When I apply the ration test to the series and simplify I get lim k...
  29. T

    What Determines the Radius of Convergence in Complex Power Series?

    Homework Statement [PLAIN]http://img153.imageshack.us/img153/4822/radiusm.jpg Homework Equations The Attempt at a Solution Using the ratio test: \left | \frac{e^{i(n+1)^2 \theta} \theta^{n+1} z^{(n+1)^2}}{e^{in^2 \theta} \theta ^n z^{n^2}} \right | = | \theta...
  30. G

    Complex Analysis: Radius of Convergence

    Homework Statement Find the radius of convergence of the power series: a) \sum z^{n!} n=0 to infinity b) \sum (n+2^{n})z^{n} n=0 to infinity Homework Equations Radius = 1/(limsup n=>infinity |cn|^1/n) The Attempt at a Solution a) Is cn in this case just 1? And plugging it in...
  31. K

    What is the Radius of Convergence for ∑[(2n+1)/2n] x^n?

    Homework Statement Find the radius of convergence of the following power series: ∑_(n=0)^∞[(2n+1)/2n] x^n) Homework Equations Ratio Test: lim_n->inf (a_n+1 / a_n) The Attempt at a Solution I got a big ugly fraction that involved both n and x
  32. L

    Finding the Radius of Convergence for (sum from n=0 to infinity)7^(-n)x^(n)

    Homework Statement Find the radius of convergence for (sum from n=0 to infinity)7^(-n)x^(n). Homework Equations The Attempt at a Solution The problem above it was a similar sum, (7^n)(x^n). That answer was that the radius of convergence was 1/7. To do this one that I posted up...
  33. J

    What Is the Radius of Convergence for This Series?

    Homework Statement find the roc of: \sum_{n=0}^{\infty}\frac{(n!)^3}{(3n)!}z^{3n} Homework Equations limsup ratio test The Attempt at a Solution i think use of limsup is quite difficult as factorial there. but i do not know how to use the ratio test because z is another...
  34. Saladsamurai

    Find Radius of Convergence from Recursion Equation

    Homework Statement Find Radius of Convergence of the corresponding power series solution from Recursion Equation alone: n^2a_{n+2} - 3(n+2)a_{n+1} +3a_{n-1} = 0 \qquad(1) Homework Equations R = 1/L where L = \lim_{n\rightarrow\infty}\left|{\frac{a_{k+1}}{a_k}\right|\qquad(2)...
  35. Saladsamurai

    Radius of Convergence for a Power Series with Ratio Test

    Homework Statement Determine the radius of Convergence using the ratio test of: \sum_o^{\infty}\frac{n^6}{3^n+n}(x+4)^{8n+1}\qquad(1) Homework Equations R = \frac{1}{\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|}\qquad(2)The Attempt at a Solution Ok. In order to use (2), we...
  36. jegues

    Radius of Convergence Problem with Solution Attempt

    Homework Statement See figure attached for problem statement as well as my attempt. Homework Equations The Attempt at a Solution See 2nd figure attached. I don't know how to rid myself of that last n! I've got kicking around in the numerator. Any ideas?
  37. P

    On the radius of convergence of a power series

    Hi, I'm new here. I am curious that why a power series must have a radius of convergence? I mean, even in a complex plane, there is always a so-called convergent radius for a power series. Is it possible that a power series is convergent for a certain range in one direction, and for an apparent...
  38. S

    Taylor series radius of convergence

    Hi, We need a generic expression of a taylor series nth term to find out the radius of convergence of the series. However, there are series where I don't think it is even possible to find a generic term. How do we find the radius of convergence in such cases? e.g. sqrt (1 - x^2) There...
  39. N

    Radius of convergence question(complex function)

    i was by my teacher that the radius of convergence is what smaller then the number which makes the denominator 0. if f(x)=\frac{1}{1-z} then the radius is 1 and because 1-1=0 so it is analitical on |z|<1 so if i apply the same logic f(x)=\frac{-2}{z-1} 1 still makes denominator 0...
  40. S

    Radius of convergence of a power series(complete result, hopefully)

    Homework Statement I have post this before but now I have come up with the complete result hopefully Anyway given the power series \sum_{j=0}^{\infty} F_{j} z^j find the radius of convergence around zero and F_j = F_{j-1} - F_{j-2} and that j \geq 2 The Attempt at a Solution...
  41. S

    Radius of Convergence of Fibonacci sequence

    Radius of Convergence of Fibonacci sequence :) Homework Statement Given the Fibonacci sequence where \frac{1}{1-x-x^2} = \sum_{n=0}^{\infty} F_{n} x^n find the radius of convergence around zero. Homework Equations Ratio test The Attempt at a Solution By the radio test...
  42. A

    Find Radius of Convergence of this Complex Series

    Hi, am a bit stuck with this. Find the radius of convergence of the complex series (Sigma n=1 to infinity) (z - e)^n! I know that the answer is R=1 but I'm not sure how to get there. It's the factorial as a power which I'm not sure about, have seen this in some other problems too. I...
  43. R

    Power Series- radius of convergence

    Homework Statement determine the radius of convergence of the given power series \sum^{inf}_{n=1}\frac{n!x^n}{n^n} Homework Equations The Attempt at a Solution I did the ratio test then I had to take the 'ln' but, my answer is this |e|<1 for the series to converge. It...
  44. F

    Finding Radius of Convergence: Ratio Test for Series

    Homework Statement find the interval of convergence of \sum[(2k+1)!/((2k)((k!)2)]* [xk] Homework Equations Ratio Test The Attempt at a Solution I already found that it converges on (-1/2, 1/2) by using power series with b=0 and testing the rest of it as ak. However, I am unsure...
  45. A

    Radius and Interval of Convergence for (3x-2)^2/n 3^n Series

    Homework Statement find the radius of convergence and interval of convergence of the series ∞ Σ (3x-2)^2 / n 3^n n=1 Homework Equations The Attempt at a Solution
  46. J

    Radius of convergence of a power series

    Homework Statement Find the radius of convergence of \sumn!*xn from n=0 to \infty Homework Equations The Attempt at a Solution I did the ratio test and was able to get it down to abs(x) * lim as n approaches \infty of abs(n+1). It seems to me that the radius of convergence...
  47. B

    Does anyone know how to find radius of convergence for sin x and e^x

    [sloved]Does anyone know how to find radius of convergence for sin x and e^x We know that to find radius of convergence we use ratio test (ie lim {a_n+1} /{a_n}) Can this method be used for sin x and e^x? ( whose radius of convergence is -infinity and infinity) if radius of convergence is...
  48. P

    What's the radius of convergence for 1/N expansion in QCD?

    I'm asking this question as someone who has not studied this topic technically. By radius of convergence, I mean exact results, not just approximations, can be obtained by summing sufficiently many terms. (does this ever happen?) I don't mind if you need 1000 terms, as long as the series is...
  49. L

    Property of radius of convergence

    I have a question regarding the radius of convergence and hopely someone can help me with it. Suppose \SigmaNANZN-1 is given and if its primitive exists, will these two polynomials have the same radius of convergence?
  50. B

    Solving the Radius of Convergence of a Periodic Power Series

    Homework Statement Consider the power series Σanxn = 1+2x+3x2+x3+2x4+3x5+x6+… in which the coefficients an=1,2,3,1,2,3,1,... are periodic of period p=3. Find the radius of convergence. Homework Equations The Attempt at a Solution My attempt at a solution was to first state...
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