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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

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

where

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


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

References

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

Mathematics Encyclopedia

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