how does one prove that 1/(e^z) = e^(-z)?
it seems like such an obvious statement.
Well it's obvious if you consider it a definition. However if you've defined as the power series then it's less obvious. It's possible to prove, using the power-series definition, that , using an essentially combinatorial argument (which I can supply if you wish). It also follows from the power-series definition that . From these two identities it follows that
So that is the multiplicative inverse of , i.e. . Note also that the above shows that is never 0, because it is always invertible.
Hope that helps!