Multi part problem to prove functional relation of the exponential function

Question

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that

$\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = 1-z+\frac{z^2}{2} - \frac{z^3}{6} + . . .$

I don't know how I can relate this to $\large\frac{1}{\exp(z)}$ though.

Answer 1

Notice that you are allowed to use the relation $\exp(z+w) = \exp(z)\exp(w)$. Use also (iii) and just notice that $$ 1 = \exp(0) = \exp(z+(-z)) = \exp(z)\exp(-z). $$