1. Limited time only! Sign up for a free 30min personal 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!

Behaviour of series (radius of convergence)

  1. Apr 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Series:
    [tex]\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n}[/tex]

    what is the behaviour of the series at radius of convergence [tex]\rho_o=-z[/tex] ?


    2. Relevant equations



    3. The attempt at a solution
    So I can specify that the series is monatonic if z is non negative as [tex]\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(-z)^n}{na^n}[/tex] right?

    But then I suppose I have to do the integral test but im a bit confused because you cannot integrate [tex](-1)^{(x+1)}\frac{(-z)^x}{xa^x}[/tex]?

    Thanks, in advance!
     

    Attached Files:

    Last edited: Apr 26, 2014
  2. jcsd
  3. Apr 26, 2014 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    It's not clear to me what the various symbols mean. What is ##\rho_0##? What is ##z##? How are they related to the series and ##x##?

    Most of the tests you have for convergence apply to positive or non-negative series. You don't have that here, do you? Or do you?
     
  4. Apr 26, 2014 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I think you should figure out what the radius of convergence ##\rho_0## of the given series is first. The ratio test should work nicely. Then substitute and simplify.
     
  5. Apr 26, 2014 #4
    Sorry for being unclear, I found the radius of convergence to be [tex]\rho_o=|a|[/tex] using the ratio test, forgot to include that in the opening post. I am unclear as to how I am supposed to figure the behaviour based on this information, the question literally gets me to find the radius of convergence from the series I mentioned and figure the behaviour of the series when [tex]z=-\rho_o[/tex]. I have added an image attachment of the question to clear things up hopefully.
     
    Last edited: Apr 26, 2014
  6. Apr 26, 2014 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I THINK they want you to substitute -|a| for x in the series and determine convergence of the resulting series. The result will depend on whether you take 'a' to positive or negative. But the series will simplify a lot in either case. Actually since you posted the original problem, it's safe to assume 'a' is positive.
     
    Last edited: Apr 26, 2014
  7. Apr 26, 2014 #6
    Yeah, that was my initial thought, just confuses me why the question uses z.. ooh well thanks I can actually get somewhere if I sub a in.
     
  8. Apr 26, 2014 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You can look at the two series obtained by setting x = a and x = -a (both of which have |x|=a---assuming a > 0).
     
  9. Apr 26, 2014 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The use of z instead of x is probably a typo. I'd ignore it. And your series is not complete. It should have an n=0 term. What is it?
     
  10. Apr 26, 2014 #9
    Thanks guys!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Behaviour of series (radius of convergence)
Loading...