Abel's Theorem

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Abel's Theorem may refer to:

Abel's Limit Theorem

Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$.


Then:

$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$

where $\ds \lim_{x \mathop \to 1^-}$ denotes the limit from the left.





Source of Name

This entry was named for Niels Henrik Abel.


Historical Note

Abel's Theorem first appears in Abel's paper Mémoire sur une Propriété Générale d'une Classe Très-Étendue de Fonctions Transcendantes.


Carl Gustav Jacob Jacobi remarked that it was the greatest discovery in integral calculus in the $19$th century.