Calculus: Analytic Geometry and Calculus, with Vectors

12.3 Alternating series and Fourier series 611

so on. Because the quantities in parentheses are positive, the formulas

(12.314) S2k = (a, - a2) + (aa - a4) + + (a2k-1 - a2k)
(k = 1,2,
show that 0 < s2 < s4 < X6 < and the formulas

(12.315) S2k-1= a1 - (a2 - as) - (a4 - a5)- - (a2k-2 - a2k-1)

(k = 1,2, )

show that a1 = s1 > Sa > Ss > S7 >. When it > k, these facts
and the formula

(12.316)

imply that

(12.317)

S2n+1 = S2n + a2,+1 > S2n

0 < S2k < S2n < S2,,+1 < S2k-1 < a1

as Figure 12.313 indicates. The bounded increasing sequence s2, S4,

S6,. and the bounded decreasing sequence sl, sa, ss, must have
limits and these limits must be equal, since S2n+1 = S2n + a2,+1 and
a2n+1 -> 0 as n - oo. Letting s be the value of these limits, we have

(12.318) lim sn = s.

n-p

In case n is even, we have sn < s < sn+1 and hence

0 <s - sn < sn+1 - sn= lan+11.

This formula and a similar one holding when it is odd give the conclusion

of Theorem 12.31.

The remainder of the text of this section gives a preview of fundamental

ideas about series that are called Fourier (1768-1830) series. While

snatches of the story can be understood by everyone, most of the results

are given without proof and it is necessary to study pure and applied

mathematics for a few years to obtain a full appreciation of the whole

story. Let L be a given positive number, and let functions 01, 2.

03,. - be defined by

(12.32) Ok(x) = .rL sin x (k = 1,2,3,. - ).

Let E be the closed interval 0 < x S L. Because the little trick enables

us to obtain formulas that have many other applications, we writefE

instead of foL. It can then be shown that

(12.33) f, 10k(x)I2 dx = 1, f, 4,j(x)4k(x) dx= 0 (9 54 k).

On this account, we say that the functions 01, q52, 0a,. constitute

an orthonormal set over E. Now let f be a function which is defined over