Irrationality of certain fast converging series and infinite products

Let (u₍) ₍₀ be an unbounded sequence of positive integers such that u₍+₁= u₍^2+O (u₍^) for some positive rational number and some ] 0, 2[. Let (r₍) ₍₀ be a sequence of rational numbers satisfying ``weak'' growth conditions. We give necessary and sufficient conditions for the series ₍=₀^r₍/u₍ and the infinite product ₍=₀^ (1+r₍/u₍) to be rational numbers. Moreover, in case of irrationality, we obtain an upper bound for their irrationality exponents.