I may have sent a bogus Mathematica 7.0 bug report. SumConvergence[(n + 1)!/((n + 2)*(n + I + 1)!), n,...] abstains for all four advertised Methods, and
HypergeometricPFQ[{1, 2, 2}, {3, I + 2}, 1] gives ComplexInfinity
Macsyma promptly says (c37) (hyper_f[3,2]([1,2,2],[3,%i+2]),%% = dfloat(%%))
(d37) hyper_f ([1, 2, 2], [3, %i + 2]) = 3, 2 0.66246689227311d0 - 1.48553272123584d0 %i
via a 3x3 matrix transformation. These terms decrease like the harmonic series, but a denominator (n+I)! provides oscillation, so I bet on Macsyma. But further analysis shows the oscillation periods increasing exponentially, which is reminiscent of the proof of harmonic divergence by taking exponentially bigger gulps, except here we have alternating "sign". I.e., the nth partial sum may go like C + A n^i, with C possibly (d37). Macsyma's formula comes from path invariant matrix products along two edges of an infinite rectangle, where the other two edges generally vanish, but maybe not in this case. Could these oscillations ever-so-gradually grow or shrink by some higher order effect of taking doubly exponentially many terms?
ComplexInfinity seems unlikely, but do I owe Wolfram's bugchasers an apology for claiming convergence?
Obviously yes.
A simpler and probably equivalent problem might be sum_n n!/(n+i+1)!, which Macsyma claims is simply -1/Gamma(i) = 0.56960764103668d0 - 1.83074439659052d0 * %i
= (1-i)/(i+1)! = :z
simply by the 2F1[1] formula, but gets the same result by the 3x3 transformation.
The latter sum telescopes, and differs from the former by a convergent series. The telescoped partial sums orbit z at eventually unit radius and exponentially increasing period.
--rwg