9. Compute
$$
\sum_{k=0}^{\infty} \frac{4}{(4 k)!}
$$
Solution: $\quad e+1 / e+2 \cos 1$
This is the power series
$$
4+\frac{4 x^{4}}{4!}+\frac{4 x^{8}}{8!}+\cdots
$$
evaluated at $x=1$. But this power series can be written as the sum
$$
\begin{aligned}
& \left(1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+\frac{x^{5}}{5!}+\frac{x^{6}}{6!}+\frac{x^{7}}{7!}+\cdots\right) \\
+ & \left(1-\frac{x}{1!}+\frac{x^{2}}{2!}-\frac{x^{3}}{3!}+\frac{x^{4}}{4!}-\frac{x^{5}}{5!}+\frac{x^{6}}{6!}-\frac{x^{7}}{7!}+\cdots\right) \\
+ & 2\left(1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots\right) \\
= & e^{x}+e^{-x}+2 \cos x .
\end{aligned}
$$

It follows that the quantity is $e+1 / e+2 \cos 1$.