Results 1 to 3 of 3

Math Help - Radius of convergence

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Radius of convergence

    Find the radius of convergence of the power series \sum a_nx^n for the following choices of (a_n; n \geq 0).

    a_n=a^{n^2} for some constant a \in (0, \infty).
    So far I have this:

    \sum a^{n^2}x^n=1+a^4x^2+a^9x^3+......

    but |a^{n^2}| \leq |a^n|

    Therefore: |\sum a^{n^2}x^n|=|1+a^4x^2+a^9x^3+... |\leq | \sum a^{n}x^n|=|1+(ax)^2+(ax)^3+...|

    This will only converge if a<1. Hence the radius of convergence of | \sum a^{n}x^n|=|1+(ax)^2+(ax)^3+...| is 1 (ie. R=1).

    So I have an upper bound, but I can't find a lower bound. Since x can be negative (i'm assuming from the question that x \in \mathbb{R}) I can't just make the original power series greater than 1!

    Any help would be great!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,
    Quote Originally Posted by Showcase_22 View Post
    but |a^{n^2}| \leq |a^n|
    This is only true if a>1 !

    Then this inequality
    Therefore: |\sum a^{n^2}x^n|=|1+a^4x^2+a^9x^3+... |\leq | \sum a^{n}x^n|=|1+(ax)^2+(ax)^3+...|
    is false
    I don't know for the rest
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    ah, okay then.

    Well how about this:

    a^{n^2}<1 for the power series to converge.

    Hence:

    n^2 ln(a)<0

    ln(a)<0

    hence a<1.

    Therefore the radius of convergence is 1.

    ie. \ R=1.

    I would have to prove that the series diverges for a \geq 1 but I think I can do that.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: March 27th 2011, 08:42 PM
  2. Replies: 1
    Last Post: May 13th 2010, 02:20 PM
  3. Replies: 2
    Last Post: May 1st 2010, 10:22 PM
  4. Replies: 1
    Last Post: November 13th 2009, 07:42 AM
  5. series convergence and radius of convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 15th 2008, 09:07 AM

Search Tags


/mathhelpforum @mathhelpforum