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...
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...
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