Answer
$$e^x\sin x
=x+x^2+\frac{x^3}{3}+...$$
Work Step by Step
By making use of Table 2, we have the Maclaurin series for $ \sin x$ and $e^x$ as follows
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$$
$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$$
Now, we have the Maclaurin series of $e^x\sin x$ as:
$$e^x\sin x=\\
=(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...)(x-\frac{x^3}{3!}+\frac{x^5}{5!}...)\\
=x+x^2+\frac{x^3}{3}+...$$
