What is the interval of convergence of ##sum (x^(2n) )/( (2n)! )##?

For all ##x in RR##

We know that

##e^x = sum_k x^k/(k!)##

and that

##e^x + e^(-x) = 2sum_k x^(2k)/((2k)!) = 2cosh(x)##

so

##sum_k x^(2k)/((2k)!) = cosh(x)## which converges

for all ##x in RR##