How is the formula for (I+N)^-1 derived when N is nilpotent?

[brydustin]

I am curious how to derive the (I+N)^-1 = I - N + N^2 - N^3 + ... N^(k-1) + 0
Where N^k = O, because we assume that N is nilpotent.

Actually I'm just supposed to show that the inverse always exists (for my homework), but I'm not asking how to find existence, I want to know how this equation is derived (assuming existence).

Thanks...

 

[micromass]

What happens if you multiply

$$(I+N)(I-N+N^2-N^3+...N^{k-1})$$

 

[brydustin]

AH! Thanks... all the "middle parts fall out

I - N + N - N^2 + N ^2 -... - N^(k-1) +N^(k+1) + N^k

and you are left with Identity, which shows that its the inverse. Thanks :)