Solution
= Solution
The <power series> for $e^{y^3}$ has <uniform convergence> on every compact interval, so <termwise integration> from zero to $x$ is valid. Thus
$$
\int_0^x e^{y^3}\,dy=\sum_{n=0}^{\infty}\frac{1}{n!}\int_0^x y^{3n}\,dy
=\boxed{\sum_{n=0}^{\infty}\frac{x^{3n+1}}{(3n+1)n!}}.
$$
This <power series> converges for every real $x$, including negative $x$ with the usual orientation of a <definite integral>.