  We prove that a probability measure on the real line has a moment of order p
(even integer), if and only if its R-transform admits a Taylor expansion with p
terms. We also prove a weaker version of this result when p is odd. Then, we
apply this to prove that a probability measure whose R-transform extends
analytically to a ball with center zero is compactly supported, and that a free
infinitely divisible distribution has a moment of order p even, if and only if
its Levy measure does so. We also prove a weaker version of the last result
when p is odd.
