试将函数f(z)=(z-1)/(z+1)按z-1展开成泰勒展式求

Question

试将函数f(z)=(z-1)/(z+1)按z-1展开成泰勒展式
求

Answer 1

记t=z-1, 则z=t+1, 将f(z)展开成t的幂级数即可.
f(z)=t/(t+2)=1-2/(t+2)=1-1/(1+t/2)
=1-[1-t/2+(t/2)^2-(t/2)^3+...]
=t/2-t^2/2^2+t^3/2^3-t^4/2^4+.