taylor series please i need immediate help

Question

Expand 1/z by Taylor series about a point z=1.

what I have done really makes no sense because I have no idea about it.

I can only think of 1/z=1/z-1+1 1/z=1/((z-1)*(1/z-1))

Answer 1

Let $f(z) = 1/z$.

The Taylor series centered at $z = 1$ is

$f(1) + f'(1)(z-1) + \frac{1}{2}f''(1)(z-1)^2 + ...$

Further

$f(1) = 1$

$f'(1) = -1/(1)^2 = -1$

$f''(1) = 2/(1)^3 = 2$

$f'''(1) = -6/(1)^4 = -6$

$f^{(n)}(1) = (-1)^n n!$