Abel theorem

Abel theorem

[′ä·bəl ′thir·əm]

(mathematics)

A theorem stating that if a power series in z converges for z = a, it converges absolutely for | z | < |="">a |.

A theorem stating that if a power series in z converges to f (z) for | z | < 1="" and="" to="">a for z = 1, then the limit of f (z) as z approaches 1 equals a.

A theorem stating that if the three series with n th term a_n, b_n, and c_n = a₀ b_n + a₁ b_n-1+ ⋯ + a_nb₀, respectively, converge, then the third series equals the product of the first two series.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.