1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Radius of convergence

  1. Jan 29, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the radius of convergence of


    3. The attempt at a solution

    i have got to lZl i (n+1)^2/3n^2

    but am unsure how to complete it...
    Last edited: Jan 29, 2008
  2. jcsd
  3. Jan 29, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    For the sum to converge, the norm of the terms must go to zero.
    So, what is:
    [tex]\lim_{n \rightarrow \infty} \left | n^2 \left(\frac{-iZ}{3}\right)^n\right|[/tex]
  4. Jan 29, 2008 #3


    User Avatar
    Science Advisor

    How and why did you "get" that? At first I thought you were using the "root test" but that won't work with the n2.

    (Am I correct that "n" is the index of summation and "i" is the complex base? If so |i|= |-1|= 1)

    I would be inclined to use the "ratio test": a sequence [itex]\sum a_n[/itex] converges if the ratio [itex]|a_{n+1}/a_n|[/itex] converges to a number less than 1. Here, [itex]|a_{n+1}|= (n+1)^2 Z^{n+1}/3^{n+1}[/itex] so the ratio becomes [itex]((n+1)/n)^2 Z/3[/itex]. Since (n+1)/n goes to 1, so does ((n+1)/n)^2 and we have |Z|/3< 1. The radius of convergence is 3.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Radius convergence Date
Complex Analysis/Radius of Convergence question. Apr 3, 2017
Radius of Convergence Dec 10, 2016
Finding the Radius of Convergence through Ratio Test Apr 23, 2016
Radius of Convergence Apr 19, 2016