approximating it by polynomials from above and below (using the Weierstrass approximation theorem and a little extra fudging) and using the fact that the coefficients an are positive. In particular the function given by g(t) = 1/t if 1/e < t < 1 and 0 otherwise has this property. But then for x = e−1/N the sum Σanxng(xn) is a0 + ... + aN, and the integral of g is 1, from which the Hardy–Littlewood theorem follows immediately. Examples Non-positive coefficients The theorem can fail without the condition that the coefficients are non-negative. For example, the function is asymptotic to 1/4(1–x) as x tends to 1, but the partial sums of its coefficients are 1, 0, 2, 0, 3, 0, 4, ... and are not asymptotic to any linear function. Littlewood's extension of Tauber's theorem In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If an = O(1/n), and as x ↑ 1 we have then This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it. Prime number theorem In 1915 Hardy and Littlewood developed a proof of the prime number theorem based on their Tauberian theorem; they proved where Λ is the von Mangoldt function, and then conclude an equivalent form of the prime number theorem. Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971. Notes External links Tauberian theorems