[math-fun] Help with a sum of a series

Hi all, I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are: 1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ... I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum? Thanks, Kerry