Editing Abelian and Tauberian theorems (section)

==Abelian theorems==

For any summation method ''L'', its '''Abelian theorem''' is the result that if ''c'' = (''c''<sub>''n''</sub>) is a [[Limit of a sequence|convergent sequence]], with [[Limit of a sequence|limit]] ''C'', then ''L''(''c'') = ''C''. {{clarify|date=March 2022}}

An example is given by the [[Cesàro mean|Cesàro method]], in which ''L'' is defined as the limit of the [[arithmetic mean]]s of the first ''N'' terms of ''c'', as ''N'' tends to infinity. One can [[mathematical proof|prove]] that if ''c'' does converge to ''C'', then so does the sequence (''d''<sub>''N''</sub>) where

: <math>d_N = \frac{c_1+c_2+\cdots+c_N} N.</math>

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 ''c''<sub>''N''</sub> 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.

The name derives from [[Abel's theorem]] on [[power series]]. In that case ''L'' is the ''radial limit'' (thought of within the [[complex plane|complex]] [[unit disk]]), where we let ''r'' tend to the limit 1 from below along the real axis in the power series with term

: ''a''<sub>''n''</sub>''z''<sup>''n''</sup>

and set ''z'' = ''r'' ·''e''<sup>''iθ''</sup>. That theorem has its main interest in the case that the power series has [[radius of convergence]] exactly 1: if the radius of convergence is greater than one, the convergence of the power series is [[uniform convergence|uniform]] for ''r'' in [0,1] so that the sum is automatically [[continuous function|continuous]] and it follows directly that the limit as ''r'' tends up to 1 is simply the sum of the ''a''<sub>''n''</sub>. When the radius is 1 the power series will have some singularity on |''z''| = 1; the assertion is that, nonetheless, if the sum of the ''a''<sub>''n''</sub> exists, it is equal to the limit over ''r''. This therefore fits exactly into the abstract picture.