lew@ihuxr.UUCP (08/12/83)
John Riordan gives my "interesting identity" as an example on page 4 of his appropriately titled book, "Combinatorial Identities". He gives the inductive proof but he skips most of the algebra. He takes things one step further. Having defined the inverse relations: a(n) = sum k=0,n of (-1)^k * C(n,k) * b(k) b(n) = sum k=0,n of (-1)^k * C(n,k) * a(k) ... he lets b(0)=0, b(k) = 1/k. This gives for a(n) the negative of the expression we just showed equal to 1 + 1/2 + ... + 1/n. The second relation then relates 1/n to a sum over sums of 1/k. He shows this directly then states, "The first derivation is somewhat simpler, thus providing a point to the Jacobian injunction 'always invert'." A point? ... oh well, he's only up to page 5! Lew Mammel, Jr. ihuxr!lew