A lot of textbooks offer a definition of the exponential function such as this:
$$\exp(x):=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$
a) Show that the given definition for $\exp$ is correct, meaning, show that the series is convergent for every $x\in \mathbb{R}$.
b) Show, with help of a geometric series, that
$$\exp(x)=\sum_{k=0}^N\frac{x^k}{k!}+R_{N+1}(x),$$
whereas
$$R_{N+1}(x)|\leq \frac{2|x|^{N+1}}{(N+1)!} \text{ for } x\in \mathbb{R} \text{ with } |x|\leq 1+\frac{N}{2}.$$
c) Show that $\exp$ solve the differential equation $\dot{x}=x$ on $\mathbb{R}$ and $\exp(0)=1$.
We started talking about functions consisting of sums (mostly series') and this is the first exercise in my textbook concerning that topic. Also, I never saw a definition of the exponential function such as this. The only thing that came to mind was that I thought about that the definition shown in b) reminds me of the taylor polynom, but I guess that doesn't have anything to with this one. To be honest I'm all lost here. I don't even know how to approach.
I would appreciate any help. I don't know how to solve this type of questions yet.
Edit:
b) $...\leq \frac{|x|^{N+1}}{(N+1)!}(\sum_{k=0}^{\infty}(\frac{x}{N+1})^k)$, right?
But applying the geometric series to it it shows that it diverges since $x$ can be greater than $1$ too, right?
And one silly question: The exercise says that I have to show
$\exp(x)=\sum_{k=0}^{N}\frac{x^k}{k!}+R_{N+1}(x)$,
I'm still not sure why one would have to do the estimates to prove that. To be honest I don't even know what there is to prove.