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.

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  1. A

    The radius of convergence of a series

    Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!
  2. A

    Finding the Radius of Convergence for Y=6x+16 - Troubleshooting and Solution

    Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...
  3. A

    Problem in finding the radius of convergence of a series

    Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!
  4. D

    Power series: radius of convergence

    ##\sum_{k=0}^\infty \frac {2^n+3^n}{4^n+5^n} x^n## in order to find the radius of convergence i apply the root test, that is ##\lim_{n \rightarrow +\infty} \sqrt [n]\frac {2^n+3^n}{4^n+5^n}## ##\lim_{n \rightarrow +\infty} \left(\frac {2^n+3^n}{4^n+5^n}\right)^\left(\frac 1 n\right)=\lim_{n...
  5. karush

    MHB 11.8.4 Find the radius of convergence and interval of convergence

    Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (1) $$a_n=\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (2) $$\left|\dfrac{a_{a+1}}{a_n}\right| =\left|\dfrac{(-1)^{n+1} x^{n+1}}{\sqrt[3]{n+1}}...
  6. J

    MHB Power Series for f(x) and Radius of Convergence

    f(x) = 4x/(x-3)^2 Find the first five non-zero terms of power series representation centered at x = 0. Also find the radius of convergence.
  7. A

    What is the radius of convergence for a series with logarithmic terms?

    Homework Statement This is from a complex analysis course: Find radius of convergence of $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ Homework Equations I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}## The Attempt at a Solution My first reaction is that this sum looks...
  8. A

    I Unable to show the radius of convergence of a numeric series

    Hi, I've computed 512 terms of a power series numerically. Below are the first 20 terms. $$ \begin{align*} w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...
  9. isukatphysics69

    Radius of convergence of the power series (2x)^n/n

    Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...
  10. Poetria

    Addition of power series and radius of convergence

    Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...
  11. S

    Ratio Test and Radius of Convergence for ∑ ((n-2)2)/n2, n=1: Homework Solution

    Homework Statement ∞ ∑ = ((n-2)2)/n2 n=1 Homework Equations The ratio test/interval of convergence The Attempt at a Solution **NOTE this is a bonus homework and I've only had internet tutorials regarding the ratio test/interval of convergence so bear with me) lim ((n-1)n+1)/(n+1)n+1 *...
  12. karush

    MHB -z.54 find the radius of convergence

    $\tiny{10.7.37}$ $\displaystyle\sum_{n=1}^{\infty} \frac{6\cdot 12 \cdot 18 \cdots 6n}{n!} x^n$ find the radius of convergence I put 6 but that wasn't the answer
  13. B

    Radius of Convergence for Ratio Test in Calculus Questions

    Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...
  14. M

    MHB Finding Radius of Convergence for Series: n/2^n and 1/(4+(-1)^n)^3n

    Hey! :o I want to find for the following series the radius of convergence and the set of $x\in \mathbb{R}$ in which the series converges. $\displaystyle{\sum_{n=0}^{\infty}\frac{n}{2^n}x^{n^2}}$ $\displaystyle{\sum_{n=0}^{\infty}\frac{1}{(4+(-1)^n)^{3n}}(x-1)^{3n}}$ I have done the...
  15. T

    MHB Ranges and Radius of convergence

    Supposing I have this expression: $$\sum_{n = 1}^{\infty} \frac{x^n}{3^n}$$ and I need to find the values for x for which this converges and the radius of convergence. I can use the radius test: $$\lim_{{n}\to{\infty}} |\frac{{x}^{(n + 1)} 3^n}{{3}^{(n + 1)} x^n}|$$ and this equals...
  16. K

    I Complex Analysis Radius of Convergence.

    Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...
  17. J

    Finding the Radius of Convergence through Ratio Test

    Homework Statement Let f(x)= (1+x)4/3 - In this question we are studying the Taylor series for f(x) about x=2. This assignment begins by having us find the first 6 terms in this Taylor series. For time, I will omit them; however, let's note that as we continuously take the derivative of this...
  18. S

    Finding the Radius of Convergence for a Complex Function.

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > How would you find the radius of convergence for the taylor expansion of: \begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation} I was thinking that you would just differentiate...
  19. ReidMerrill

    Radius of Convergence for Σ6n(x-5)n(n+1)/(n+11) Series | Solve for x

    Homework Statement Find all values of x such that the given series would converge Σ6n(x-5)n(n+1)/(n+11) Homework EquationsThe Attempt at a Solution By doing the ratio test I found that lim 6n(x-5)n(n+1)/(n+11) * (n+12)/[6n+1(x-5)n+1(n+2)] n→inf equals 1/(6(x-5)) * lim...
  20. JulienB

    Understanding Radius of Convergence in Power Series Calculations

    Homework Statement Hi everybody! I'm a little struggling to fully understand the idea of radius of convergence of a function, can somebody help me a little? Are some examples I found in old exams at my university: Calculate the radius of convergence of the following power series: a)...
  21. G

    MHB Radius of Convergence for $\sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$

    Radius of convergence of $\displaystyle \sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$. If I let $z^2 = x$ I get a series whose radius of convergence is $2$ (by the ratio test). How do I get from this that the original series has a radius of convergence equal to $\sqrt{2}$?
  22. G

    Proving Radius of Convergence |z_0| = R for Power Series

    Homework Statement Given the power serie ##\sum_{n\ge 0} a_n z^n##, with radius of convergence ##R##, if there exists a complex number ##z_0## such that the the serie is semi-convergent at ##z_0##, show that ##R = |z_0|##. Homework EquationsThe Attempt at a Solution Firstly, since...
  23. R

    Power series where radius of convergence > lower limit

    Homework Statement Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha = \limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...
  24. R

    Finding the radius of convergence of a power series

    Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...
  25. B

    MHB Problem evaluating the limit to find the radius of convergence

    Hi everyone, I am trying to evaluate the radius of convergence for the following power series: (k!(x-1)k)/((2k)(kk)) I have begun by trying to compute L = lim k-->inf (an+1/an). To then be able to say R = 1/L. So far i have L = lim k--> inf (kk(k+1)!)/(2(k+1)k+1k!) From here i am having...
  26. B

    MHB Radius of Convergence: Evaluate & Ignoring Extra Vars

    I am attempting to evaluate the radius of convergence for a series that goes from k=0 to infinity. The series is given by (k*x^k)/(3^k). I have begun by using the ratio test and have gotten to the point L = (k+1)*x/3k Now i know i can find out the radius of convergence by simply saying R =...
  27. evinda

    MHB Understanding the Radius of Convergence of e^x and its Series Expansion

    Hello! (Wave) $$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$ i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$. Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$? Do...
  28. Shackleford

    What is the radius of convergence of

    Homework Statement z ∈ ℂ What is the radius of convergence of (n=0 to ∞) Σ anzn? Homework Equations I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences. a_n = \frac{n+(-1)^n}{n^2} limn→∞ |an|1/n The Attempt at a Solution I think that the radius of...
  29. C

    Ratio Test Radius of Convergence

    Homework Statement ∑ x2n / n! The limits of the sum go from n = 0 to n = infinity Homework EquationsThe Attempt at a Solution So I take the limit as n approaches infinity of aa+1 / an. So that gives me: ((x2n+2) * (n!)) / ((x2n) * (n + 1)!) Canceling everything out gives me x2 / (n + 1)...
  30. M

    Radius of convergence derivation

    Hi, I am likely just missing something fundamental here, but I recently just revisited series and am looking over some notes. In my notes, I have written that if ## \lim_{x \to +\infty} \frac{a_{n+1}}{a_n} = L ## Then ## | x - x_o | = 1/L ## But shouldn't the correct expression be $$ | x -...
  31. Julio1

    MHB What is the Radius of Convergence for the Power Series?

    Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$
  32. I

    MHB Taylor Series Expansion and Radius of Convergence for $f(x)=x^4-3x^2+1$

    find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?
  33. M

    Finding the Radius of Convergence for a Series with a Real Coefficient

    Hello. How do I find the radius of convergence for this problem? ##\alpha## is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$ I understand that we can use the ratio test to find R. And by using ratio test, I got R=1. But in the...
  34. A

    MHB What is the Radius of Convergence for a Series with a Real Non-Zero Alpha?

    Hello. How do I find the radius of convergence for this problem? $\alpha$ is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$
  35. M

    Macluarin series and radius of convergence

    Hello. I am stuck on this question. I'd appreciate if anyone could help me on how to do this. The question: Expand the following into maclaurin series and find its radius of convergence. $$\frac{2-z}{(1-z)^2}$$ I know that we can use geometric series as geometric series is generally...
  36. M

    Finding the Radius of Convergence for $zsin(z^2)$ in Maclaurin Series

    Hello. I need explanation on why the answer for this problem is R = ∞. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$...
  37. A

    MHB What is the Radius of Convergence for zsin(z^2)?

    Hello. I need explanation on why the answer for this problem is $R=\infty$. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$ Divide...
  38. L

    Behaviour of series (radius of convergence)

    Homework Statement Series: \sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n} what is the behaviour of the series at radius of convergence \rho_o=-z ? Homework Equations The Attempt at a Solution So I can specify that the series is monatonic if z is non negative as...
  39. L

    Radius of convergence log(a + x)

    Homework Statement determine the radius of convergence of the series expansion of log(a + x) around x = 0 Homework Equations The Attempt at a Solution So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the...
  40. H

    Ratio test for finding radius of convergence

    Homework Statement I've found that the typical way for using ratio test is to find the limit of an+1/an However, my tutor said that radius of convergence can be found by finding the limit of an/an+1 and the x term is excluded. For example:Finding the interval of convergence of n!xn/nn my...
  41. E

    Find radius of convergence and interval of convergence for the series

    x^n/(2n-1) is the series. It starts at 1 and goes to infinity. I did the ratio test on it and got abs.(x) So the radius of convergence=1, and then I plugged -1 and 1 into the original series and got that they both converged. But the answer is [-1,1). Why aren't they both hard brackets?
  42. Fernando Revilla

    MHB Binomial series (radius of convergence)

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  43. Fernando Revilla

    MHB Emil's question at Yahoo Answers (Radius of convergence)

    Here is the question: Here is a link to the question: How to find Radius of Convergence for Sum of ((x-3)^n)/(n3^n) from n =1 to inf? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  44. A

    Radius of convergence of power series

    Homework Statement The coefficients of the power series \sum_{n=0}^{∞}a_{n}(x-2)^{n} satisfy a_{0} = 5 and a_{n} = (\frac{2n+1}{3n-1})a_{n-1} for all n ≥ 1 . The radius of convergence of the series is: (a) 0 (b) \frac{2}{3} (c) \frac{3}{2} (d) 2 (e) infinite Homework EquationsThe Attempt at...
  45. Fernando Revilla

    MHB Theodore K's question at Yahoo Answers (Radius of convergence)

    Here is the question: Here is a link to the question: Calculus Power Series/Radius of Convergence/Interval of Convergence Question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  46. X

    Radius of Convergence Power Series

    Homework Statement Determine the radius of convergence and the interval of convergence og the folling power series: n=0 to infinity Ʃ=\frac{(2x-3)^{n}}{ln(2n+3)} Homework Equations Ratio Test The Attempt at a Solution Well I started with the ratio test but I have no clue where...
  47. B

    Radius of convergence and 2^1/2

    Homework Statement Suppose c_n is the digit in the nth place of the decimal expansion of 2^1/2. Prove that the radius of convergence of \sum{c_n x^n} is equal to 1. Homework Equations The Attempt at a Solution What I want to show is that limsup |c_n|^1/n = 1. Clearly for any...
  48. P

    Determining the radius of convergence

    1. Determine the raius of convergence and interval of convergence of the power series \sum from n=1 to \infty (3+(-1)n)nxn. 2. Usually when finding the radius of convergence of a power series I start off by using the ratio test: limn\rightarrow∞|((3+(-1)n+1)n+1xn+1/ (3+(-1)n)nxn| But...
  49. 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
  50. M

    Radius of convergence problem

    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...
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