Results 1 to 5 of 5

Math Help - absolute convergence

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    39

    absolute convergence

    I need to show Sum(I-T)^k from k=0 to infinity converges absolutely to T^(-1).
    Where ||I-T|| < 1.

    so I can show the inverse exists, and I know this is a geometric series but when I take the limit of (I-T)^k as k-> infinity -> I get 0...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by CarmineCortez View Post
    I need to show Sum(I-T)^k from k=0 to infinity converges absolutely to T^(-1).
    Where ||I-T|| < 1.

    so I can show the inverse exists, and I know this is a geometric series but when I take the limit of (I-T)^k as k-> infinity -> I get 0...

    Hint: \sum\limits_{n=0}^\infty r^n=\frac{1}{1-r}\,,\,\,for\,\,\,|r|<1\,,\,\,r\in\mathbb{C}

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2008
    Posts
    39
    Quote Originally Posted by tonio View Post
    Hint: \sum\limits_{n=0}^\infty r^n=\frac{1}{1-r}\,,\,\,for\,\,\,|r|<1\,,\,\,r\in\mathbb{C}

    Tonio
    But then it is 1/(matrix)...does that have a meaning
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,977
    Thanks
    1643
    Quote Originally Posted by CarmineCortez View Post
    But then it is 1/(matrix)...does that have a meaning
    When did you tell us that this problem involved matrices? In any case, (1/T)(T)= 1 so 1 over a matrix is the inverse matrix- the multiplicative inverse.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by CarmineCortez View Post
    But then it is 1/(matrix)...does that have a meaning

    I had a feeling that T must be a matrix but you never pointed out this; anyway, if you're asking this question then you know there's a definition and a meaning to "convergence of matrices, their limits and etc."

    In this case, it turns out that T\cdot\sum\limits_{k=0}^\infty (1-T)^k=I

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Absolute Convergence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 13th 2010, 07:46 AM
  2. absolute convergence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 8th 2009, 06:08 AM
  3. Absolute convergence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 17th 2008, 11:41 PM
  4. Absolute Convergence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 6th 2008, 10:50 PM
  5. help with sum for absolute convergence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 17th 2007, 10:32 PM

Search Tags


/mathhelpforum @mathhelpforum