Hardy–Littlewood Tauberian theorem

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 the sequence ${\displaystyle a_{n}\geq 0}$ is such that there is an asymptotic equivalence

${\displaystyle \sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0}$

then there is also an asymptotic equivalence

${\displaystyle \sum _{k=0}^{n}a_{k}\sim n}$

as ${\displaystyle n\to \infty }$. 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.^: 226 In 1930, Jovan Karamata gave a new and much simpler proof.^: 226