Every power series has a convergence radius R, where Sum[a_{n}x^{n}] converges if |x| < R.

The first summation is the power series equal to e^{z}

and we have a_{n} = 1/n!

lim |a_{n+1}| / |a^{n}| = 1 / (n + 1)

= 0. So our convergence radius R is infinity and the power series converges for all z.

The second summation is the power series equal to cos(z) and we know:

cos(z) = (1/2)*(e^{iz} + e^{-iz})

From the first series, we know the e^{z} is convergent on all z, so cos(z) is also convergent on all z.