Fine Art


In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.

Let {fn} be a sequence of real numbers such that fn ≥ fn+1 > 0 for n = 1, 2, …, and let {an} be a sequence of real or complex numbers. Then

\( \left |\sum_{n=1}^m a_n f_n \right | \le A_m f_1, \)


\( A_m=\operatorname{max}\left \lbrace |a_1|,|a_1+a_2|,\dots,|a_1+a_2+\cdots+a_m| \right \rbrace. \)

The inequality also holds for infinite series, in the limit as \( m \rightarrow \infty \) , if \( \lim_{m \rightarrow \infty} A_m\ \) exists.


Weisstein, Eric W., "Abel's inequality" from MathWorld.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World