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 a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equivalence ∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\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 ∑ k = 0 n a k ∼ n {\displaystyle \sum _{k=0}^{n}a_{k}\sim n} as n → ∞ {\displaystyle n\to \infty } .

Source: Wikipedia — Hardy–Littlewood Tauberian theorem (CC BY-SA 4.0)

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 a n ≥ 0 {\displaystyle a_{n}\geq 0} is such that there is an asymptotic equivalence ∑ n = 0 ∞ a n e − n y ∼ 1 y as y ↓ 0 {\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 ∑ k = 0 n a k ∼ n {\displaystyle \sum _{k=0}^{n}a_{k}\sim n} as n → ∞ {\displaystyle n\to \infty } .

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Source: Wikipedia "Hardy–Littlewood Tauberian theorem" · CC BY-SA 4.0

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