TITLE: Ring of formal power series
QUESTION [6 upvotes]: Let $$\mathbb{C}[[x]] := \left\{ \sum_{n\geq 0} a_n x^n \;\colon\; a_n \in \mathbb{C} \right\}$$ be the set of formal power series of $x$ and $$F(x) = \sum_{n\geq 0} a_n x^n \in \mathbb{C}[[x]], \; F(0) = a_0 = 1.$$

Exercise
i)  Reason that the ring of formal power series of $F'(x) = F(x)$ leads to $a_n = \frac{1}{n!}$, $n\geq 0$ and $F(x) = \exp(x)$.
ii) Prove that $F(x)$ has a multiplicative inverse in $\mathbb{C}[[x]] \Leftrightarrow a_0 \neq 0$.


i) Do you have any hint for me how to approach this one best?
ii) I searched our lecture but didn't find any further information about the ring of formal power series. What is it's multiplicative identity?
As I understand the definition of rings there is a multiplicative identity $1$ with $x \cdot 1 = 1 \cdot x = x$ and the multiplicative inverse $x'$ of $x$ is defined by $x \cdot x' = x' \cdot x = 1$.
Where is my error in reasoning or what is that multiplicative inverse?
Thank you in advance!

REPLY [4 votes]: HINT $\:$ for ii) $\:$  Using your previous exercise on composition of formal power series, you can reduce computation of general inverses to that of $\rm\displaystyle \frac{1}{1+x}\ $ by using $\rm\displaystyle\ \frac{1}{c+x\:f}\ =\ \frac{1}c\ \frac{1}{1+x\:f/c}\ $ i.e. substitute $\rm\ xf/c\ $ for $\rm\:x\:$ in the power series for $\rm\displaystyle\ \frac{1}{1+x}$