I am stuck on the following task:

Fix a vector $y \in C^n$ and let $f(x,y)$ be a function defined on $C^n \times C^n$ such that $f(0,y)=1$ and its $n$th partial derivative on $x$ satisfies $\frac{\partial^n{f}(0,y)}{\partial{x}^n}=1.$ Show that $f$ has an expension entiere series on $x$.
In addition, have we the following $$\parallel f(x,y)\parallel \leq \exp(\parallel x \parallel \times \parallel y \parallel)$$?

Any help please and thanks in advance?