Abel's Theorem

Disambiguation

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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.

This article is complete as far as it goes, but it could do with expansion. In particular: other theorems with the same name as they appear in our field of vision You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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.

This article, or a section of it, needs explaining. In particular: It is not certain which of the various Abel's Theorems are being described in Calculus Gems. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.