1. The problem statement, all variables and given/known data [n=0 to ∞] ∑x2n/n! 2. Relevant equations The ratio test 3. The attempt at a solution So, here is the ratio of of the (n+1)th term to the nth term: | [x2(n+1)/(n+1)!] / [x2n/n!] | = | [x2nx2/(n+1)!] * [n!/x2n] | = | x2/n |, which will be will be less than one when n is big enough. So this converges for all x, right?