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...

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- Jan 28th 2010, 12:56 PMCarmineCortezabsolute 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... - Jan 28th 2010, 06:25 PMtonio
- Jan 28th 2010, 06:50 PMCarmineCortez
- Jan 29th 2010, 01:53 AMHallsofIvy
- Jan 29th 2010, 02:58 AMtonio

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 $\displaystyle T\cdot\sum\limits_{k=0}^\infty (1-T)^k=I$

Tonio