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

References

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