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Cauchy-Schwarz Sum Inequality

\vert{\bf a}\cdot {\bf b}\vert \leq \vert{\bf a}\vert\, \vert{\bf b}\vert.

\left({\,\sum_{k=1}^n a_kb_k}\right)^2\leq \left({\,\sum_{k=1}^n {a_k}^2}\right)\left({\,\sum_{k=1}^n {b_k}^2}\right).

Equality holds Iff the sequences $a_1$, $a_2$, ... and $b_1$, $b_2$, ... are proportional.

See also Fibonacci Identity


Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1092, 1979.

© 1996-9 Eric W. Weisstein