TITLE: Two random variables with same moments
QUESTION [7 upvotes]: Reading http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter10.pdf pages 368-370. it states "if we delete the hypothesis that have finite range in the above theorem, then the conclusion is no longer necessarily true". 
So my question is 

Do you know a (simple?) example of two discrete random variables with
  finite moments equal for each order but different distributions?

REPLY [5 votes]: Not quite what you're asking for, but a standard example of continuous random variables with the same moments yet distinct distributions is
$f_1(x) = e^{- (\log{x})^2/
2} \,
/ \,(x\sqrt{2\pi})$
$f_2(x) =
f_1(x)(1 + \sin(2
\pi
\log{x})/2)$
where $ x > 0$.