We present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) = T1(n+1) - T1(n) + T2(n), and T2 is minimal in some sense. This solves the decomposition problem for indefinite sums of hypergeometric terms: T1(n+1) - T1(n) is the ''summable part'' and T2(n) the :non-summable part'' of T(n).