Prove $\sum \frac{\cos nz}{n!}$ converges on compact sets Prove that
$$\sum_{n=1}^{\infty} \displaystyle\frac{\cos nz}{n!}$$
Is an entire function, which means that it uniformly convergent on compact sets

Sketch: 
You can use that $\cos(nx + n iy) = \cos nx \cosh ny - i\sin nx \sinh ny$ to obtain that  $|\cos nz| \leq e^{|ny|}$. Since factorials grow faster than exponentials, the complex version of the Weierstrass $M$-test will imply the result.
