Hardy–Littlewood Tauberian theorem
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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 is such that there is an asymptotic equivalence
then there is also an asymptotic equivalence
as . 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.[1]: 226 In 1930, Jovan Karamata gave a new and much simpler proof.[1]: 226