Abelian and Tauberian theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In the theory of integral transforms, Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely, Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function. For any summation method L, its Abelian theorem is the result that if c = (cn) is a convergent sequence, with limit C, then L(c) = C. An example is given by the Cesàro method, in which L is defined as the limit of the arithmetic means of the first N terms of c, as N tends to infinity. One can prove that if c does converge to C, then so does the sequence (dN) where To see that, subtract C everywhere to reduce to the case C = 0. Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take N large enough to make the initial segment of terms up to cN average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded.