500 Chapter 8 • Infinite Series of Real Numberslinear algebra course, we omit them here. Because of property 4, given any
J! E JRn, the square root vr · J! is a real number. In JR^2 and JR^3 , this quantity
represents the length of the vector r. So we generalize to JRn and call vr · J!
the length of any given vector J! in JRn.
One property of the dot product is so important to us that we prove it here
as our next theorem.Theorem 8.5.14 (Cauchy-Schwarz Inequality) l:/J!, '[/ E JRn,1--r ·vi:::; v--r · --r,;v-:v.
(The absolute value of the dot product of two n-vectors is less than or equal to
the product of their lengths.)Proof. Let J!, '[/ E JRn. The algebraic properties of the dot product in
]Rn assure us that \fr E JR,(rJ! + '[/) · (rJ! + '[/) ;::: 0
(rJ! · rJ!) + ('[/ · rJ!) + (rJ! · '[/) + ('[/ · '[/) ;::: 0
r^2 (r. r) + 2r(r. '[/) + ('[/. '[/) ;::: 0.
The left side of this inequality can be regarded as a quadratic expression
in the variable r. Since this quadratic is always ;::: 0, its discriminant must be
:::; o.
That is,
[2(r. Tf)J^2 - 4(r. r)(Tf. Tf):::; o
i.e., 4(r. T/)^2 :::; 4(r. r)(Tf. Tf).
Dividing out the 4's and taking the square root of both sides, we have the
desired inequality. •
The reader may wonder what relevance the dot product in JRn has to the
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study of the convergence of L akbk. The relevance becomes clear when we
k=l
think of this series of products as a kind of "dot product" of infinite sequences:
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L akbk = (a1, a2, a3, · ··,an,··· ) · (b1, b2, b3, · · · , bn, · · · ). (19)
k=lWe shall find that this approach is exactly what we need, especially as it
allows us to use the power of the Cauchy-Schwarz inequality.