Solution
= Solution
The coefficients in the <Taylor series> are
$$
\boxed{e^x=\sum_{n=0}^\infty\frac{x^n}{n!},\qquad e^{2x}=\sum_{n=0}^\infty\frac{2^nx^n}{n!}.}
$$
For either series, the <ratio test> gives absolute term ratio $|x|/(n+1)$ or $2|x|/(n+1)$, tending to zero for each fixed real or complex $x$. Both have infinite <radius of convergence>.