In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if, as y ↓ 0, the non-negative sequence an is such that there is an asymptotic equivalence then there is also an asymptotic equivalence as n → ∞. The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform. The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood. In 1930, Jovan Karamata gave a new and much simpler proof. Statement of the theorem Series formulation This formulation is from Titchmarsh. Suppose an ≥ 0 for all n, and as x ↑ 1 we have Then as n goes to ∞ we have The theorem is sometimes quoted in equivalent forms, where instead of requiring an ≥ 0, we require an = O(1), or we require an ≥ −K for some constant K. The theorem is sometimes quoted in another equivalent formulation (through the change of variable x = 1/ey ). If, as y ↓ 0, then Integral formulation The following more general formulation is from Feller. Consider a real-valued function F : [0,∞) → R of bounded variation. The Laplace–Stieltjes transform of F is defined by the Stieltjes integral The theorem relates the asymptotics of ω with those of F in the following way. If ρ is a non-negative real number, then the following statements are equivalent Here Γ denotes the Gamma function. One obtains the theorem for series as a special case by taking ρ = 1 and F(t) to be a piecewise constant function with value between t = n and t = n + 1. A slight improvement is possible. According to the definition of a slowly varying function, L(x) is slow varying at infinity iff for every positive t. Let L be a function slowly varying at infinity and ρ a non-negative real number. Then the following statements are equivalent Karamata's proof found a short proof of the theorem by considering the functions g such that An easy calculation shows that all monomials g(x) = xk have this property, and therefore so do all polynomials g. This can be extended to a function g with simple (step) discontinuities by