Solution
= Solution
For $F(x)=e^{e^x}$, the <chain rule> gives $F'=e^xe^{e^x}$ and $F''=(e^x+e^{2x})e^{e^x}$. Thus $F(0)=e$, $F'(0)=e$ and $F''(0)=2e$, yielding
$$
\boxed{e^{e^x}=e+ex+ex^2+O(x^3).}
$$
= Solution
For $F(x)=e^{e^x}$, the <chain rule> gives $F'=e^xe^{e^x}$ and $F''=(e^x+e^{2x})e^{e^x}$. Thus $F(0)=e$, $F'(0)=e$ and $F''(0)=2e$, yielding
$$
\boxed{e^{e^x}=e+ex+ex^2+O(x^3).}
$$