Solution
= Solution
Multiply the standard <Taylor series> through quadratic order:
$$
e^x\log(1+x)=\left(1+x+\frac{x^2}{2}+\cdots\right)
\left(x-\frac{x^2}{2}+\cdots\right).
$$
The linear term is $x$ and the quadratic coefficient is $1-1/2$, hence
$$
\boxed{f(x)=x+\frac{x^2}{2}+O(x^3)}.
$$