Solution
= Solution
Multiply the <Maclaurin series> for the exponential and the <sine>, keeping terms through degree three:
$$
e^x\sin x=\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+O(x^4)\right)
\left(x-\frac{x^3}{6}+O(x^5)\right).
$$
Only $x$, $x^2$, and $(1/2-1/6)x^3$ contribute at these orders, so
$$
\boxed{e^x\sin x=x+x^2+\frac{x^3}{3}+O(x^4).}
$$