Abel sum

== English == === Etymology === After Norwegian mathematician Niels Henrik Abel (1802-1829). === Noun === Abel sum (plural Abel sums) (mathematical analysis) Given a power series f ( x ) = ∑ n = 0 ∞ a n x n {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}} that is convergent for real x in the open interval (0, 1), the value lim x → 1 − ∑ n = 0 ∞ a n x n {\displaystyle \lim _{x\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}x^{n}} , which is assigned to f ( 1 ) = ∑ n = 0 ∞ a n {\displaystyle f(1)=\sum _{n=0}^{\infty }a_{n}} by the Abel summation method (or A-method). 1967, Jan Mikusiński, Operational Calculus, Cambridge University Press, page 102, The Abel sum of ∑ a n {\displaystyle \textstyle \sum a_{n}} is defined as the limit of the corresponding power series: lim x → 1 − 0 ∑ n = 0 ∞ a n x n {\displaystyle \lim _{x\rightarrow 1-0}\sum _{n=0}^{\infty }a_{n}x^{n}} . The existence of the Abel sum is ascertained when the series in question is known to be summable (C, r) for some value of r. ==== Derived terms ==== Abel summable Abel summation ==== Related terms ==== Abel mean Abel summation method === See also === summability method summation method === Further reading === Divergent series on Wikipedia.Wikipedia Abel's theorem on Wikipedia.Wikipedia Abel's summation formula on Wikipedia.Wikipedia === Anagrams === bemauls, malesub