1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    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!

Determining the radius of convergence

  1. Dec 31, 2012 #1
    1. Determine the raius of convergence and interval of convergence of the power series [itex]\sum[/itex] from n=1 to [itex]\infty[/itex] (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[itex]\rightarrow[/itex]∞|((3+(-1)n+1)n+1xn+1/ (3+(-1)n)nxn|

    But this limit does not exist since this equation is just oscillating between 0 and +infinity.

    Is there a radius of convergence?
     
    Last edited: Dec 31, 2012
  2. jcsd
  3. Dec 31, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Every power series has a radius of convergence- but it is possible that the radius of convergence is 0 or infinity.

    Here, you have
    [tex]\frac{(3+(-1)^{n+1})^{n+1}}{(3+(-1)^n)^n}|x|[/tex]

    The simplest way to handle this is to look at n even and odd separately:
    1) if n is even, then n+1 is odd so [itex](-1)^n= 1[/itex] and [itex](-1)^{n+1}= -1[/itex] so that we have
    [tex]\frac{(2^{n+1}}{4^n}|x|= (2)(1/2)^n|x|[/tex]
    and, as n goes to infinity that goes to 0.

    2) if n is odd, then n+1 is even so [itex](-1)^n= -1[/itex] and [itex](-1)^a{n+1}= 1[/itex] so that we have
    [tex]\frac{4^{n+1}}{2^n}|x|= 4(2^n)|x|[/tex]
    which does not converge. The series converges only for x= 0 and the radius of convergence is 0.
     
  4. Dec 31, 2012 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You've only shown that the limit in the ratio test doesn't exist. That means the ratio test is inconclusive. You can't say anything about the radius of convergence from that. You want to split the series into two other series consisting of even and odd terms of the original series. Those both have a well defined radius of convergence.
     
  5. Dec 31, 2012 #4
    I split the original series into even and odd terms and got the radius of convergence =1/4 for even terms and =1/2 for odd terms. Is this correct?
     
  6. Dec 31, 2012 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, so what do you say for the radius of convergence of the whole series?
     
  7. Dec 31, 2012 #6
  8. Dec 31, 2012 #7

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Think about it again. One of the series diverges for |x|>1/4.
     
  9. Dec 31, 2012 #8
    Oh so it will the orginal series will have a radius of convergence of 1/4?!
     
  10. Dec 31, 2012 #9

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it will be the smaller of the two radii.
     
  11. Dec 31, 2012 #10
    Does that mean the interval of convergence is (-1/4,1/4)? Because when I let x=1/4,=-1/4 I get 1 and +/-1 respectively which are divergent series. Therefore it is absolutely convergent in (-1/4,1/4) and is divergent everywhere else.

    THank you for your help on this.
     
  12. Dec 31, 2012 #11

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's it. You're welcome.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Determining the radius of convergence
  1. Radius of Convergence (Replies: 4)

  2. Radius of Convergence (Replies: 4)

  3. Radius of convergence (Replies: 2)

  4. Radius of convergence (Replies: 7)

  5. Radius of convergence (Replies: 6)

Loading...