Results 1 to 3 of 3

Math Help - Formal power series: inverse element

  1. #1
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
    From
    Brugge, BEL
    Posts
    39

    Formal power series: inverse element

    Hello,
    the problem is to calculate inverse element (reciprocal) of 1-x^3 in ring $\mathbb{R}[[x]].

    I skimmed some Wikipedia and found following formula:
    Inverting series

    But I don't know to use it. Also the following model answer didn't help me either:
    "Task is to calculate inverse element (reciprocal) of 1-x in the ring $\mathbb{R}[[x]].

    So, we have to calculate A(x)=a_0+a_1x+a_2x^2+..., that
    (1-x)(a_0+a_1x+a_2x^2+...)=1.

    When the a_0 is solved, and then a_1..., we see that inverse element is 1+x+x^2+..."

    I don't understand how those a_0 etc. are solved. Only thing, that is clear to me is, that 1+x+x^2+... is generated by \frac{1}{1-x}, which equals (1-x)^{-1}.

    So, any help is appreciated. Thanks very much!
    Last edited by Greg98; February 14th 2011 at 09:39 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    The formal series S(x)=\sum_{n\geq 0}a_nx^n\in\mathbb{R}[[x]] has inverse element iff a_0\neq 0 .

    If T(x)=\sum_{n\geq 0}b_nx^n is the inverse of the formal series 1-x^3 then, the equality (1-x^3)T(x)=1 will provide:

    \begin{Bmatrix}b_0=1\\b_1=0\\b_2=0\\b_3-b_0=1\\b_n-b_{n-3}=0\;(n\geq 4)\end{matrix}

    So, you can determine T(x) .


    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466
    Another way to do this is to think of \frac{1}{1- x^3} as the sum of a geometric series.

    \sum_{n=0}^\infty r^n= \frac{1}{1- r}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Formal power series: identity problem
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 16th 2011, 01:29 PM
  2. Formal Power Series
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: February 14th 2010, 08:45 PM
  3. Replies: 2
    Last Post: September 13th 2009, 06:27 AM
  4. Formal power series & Taylor series
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 19th 2009, 09:01 AM
  5. inverse tan x power series
    Posted in the Calculus Forum
    Replies: 5
    Last Post: August 28th 2006, 01:26 PM

Search Tags


/mathhelpforum @mathhelpforum