Results 1 to 5 of 5

Math Help - Power Series - Convergence and Sum

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    128

    Power Series - Convergence and Sum

    Consider power series

    \sum a_{n}x^n = 1 + 2x + 3x^2 + x^3 + 2x^4 + 3x^5 + x^6 ...

    in which coefficients an = 1, 2, 3, 1, 2, 3, 1 ... are periodic of period 3. Find the radius of convergence and the sum of this power series.

    For the radius, is it simply a task of doing the limit test, where \lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Lord Darkin View Post
    Consider power series

    \sum a_{n}x^n = 1 + 2x + 3x^2 + x^3 + 2x^4 + 3x^5 + x^6 ...

    in which coefficients an = 1, 2, 3, 1, 2, 3, 1 ... are periodic of period 3. Find the radius of convergence and the sum of this power series.

    For the radius, is it simply a task of doing the limit test, where \lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}?
    Note that for positive x we have that \sum_{n\in\mathbb{N}}a_nx^n\leqslant 3\sum_{n\in\mathbb{N}}x^n
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    128
    How do I apply what you said into this problem?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Lord Darkin View Post
    How do I apply what you said into this problem?
    Figure it out
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2009
    Posts
    128
    \lim a_{n+1} * x < 1

    \lim x < \frac{1}{a_{n+1}}

    x = \frac{1}{3}, \frac{3}{2}, 2
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. convergence of a power series
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: August 22nd 2011, 05:59 AM
  2. Power series convergence
    Posted in the Calculus Forum
    Replies: 9
    Last Post: April 29th 2011, 10:46 AM
  3. Convergence of power series
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 29th 2011, 01:18 AM
  4. Power Series Convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 3rd 2009, 07:57 PM
  5. Power series convergence
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 17th 2007, 01:25 PM

Search Tags


/mathhelpforum @mathhelpforum