Taylor expansion of $\exp(\sin x)$ around $c=0$ 
Consider the Taylor's expansion around $c=0$
  and find the first 4 terms for the function $\exp(\sin x)$.

I have done this but I'm not sure if is correct.
$$e^{\sin x}=\left(1+x+\frac{x^2}{2!}+\dots+\frac{x^{n-1}}{(n-1)!}\right)^{x-\frac{x^3}{3!}+\frac{x^5}{5!}+\dots+(-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!}}$$

We may avoid the use of the definition, by taking the composition of the expansions of $e^x$ and $\sin(x)$ at $x=0$:
$$\begin{align}
e^{\sin(x)}&=e^{x-\frac{x^3}{3!}+o(x^4)}\\
&=1+\left(x-\frac{x^3}{3!}+o(x^4)\right)
+\frac{1}{2!}\left(x-\frac{x^3}{3!}+o(x^4)\right)^2
\\&\qquad+\frac{1}{3!}\left(x+o(x^2)\right)^3+\frac{1}{4!}\left(x+o(x^2)\right)^4+o(x^4)\\
&=1+x-\frac{x^3}{6}+\frac{x^2}{2}-\frac{x^4}{6}+\frac{x^3}{6}
+\frac{x^4}{24}+o(x^4)\\
&=1+x+\frac{x^2}{2}-\frac{x^4}{8}+o(x^4).
\end{align}$$
