Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 619

is'. alid Ns henever the series on the right is convergent. From this it follows that

(5) limF(x - t) - 2F(x) + F(x + t)= f(x)

two t2

for each x for which the series in (1) is convergent. We conclude with a brief
outline of the sophisticated steps by which this sophisticated result is used to
prove the following difficult theorem. If the series in (1) converges to 0 for each
x, then I = Bk = 0 for each k. The present hypotheses imply that F is con-
tinuous and that the left member of (5), the generalized second derivative of F
at x, is 0 for each x. These facts can be used to prove that F must be a linear
function, and further arguments involving (2) can be used to prove that F(r) = 0
for each x. Since F(0) = F(27r) = 0, use of (2) shows that 1Io = 0. Further
arguments involving (2) show that Ak = Bk = 0 for each k. The theorem is
called a uniqueness theorem because it implies that if f is a giNen function, then
there can be at most one collection of constants -40, 41, 4,, and B1, B-,,
for which the series in (1) converges to f(x) for each x. This means that if the
formulas (1) and

(6) f(x) = Co + (Ck cos kx + Dk sin kx)

k=1

are both valid for each x, then Ck = 1k and Dk = Bk for each k. Our brief
glimpse of the Riemann theory of trigonometric series can make us aware of the
fact that the uniqueness theorem for trigonometric series has been proved, and
the proof involves mathematical ideas that we have not yet assimilated. This
matter is important, because trigonometric series appear even in quite elementary
applied mathematics and we need some authoritative information to help us
appraise the revelations appearing in textbooks that give superficial treatments
of the subject.

12.4 Power series The series

(12.41) co + ci(x - a) + c2(x - a)2 + c3(x - a)3 +

in which a and co, e1, e2, .. are constants, is called a power series in

(x - a). The fundamental reason for importance of these things lies

in the fact that powers of (x - a) are relatively easy to calculate, to
differentiate, and to integrate. Some power series, like the series

(12.411) 0!+1!(x-a)+2!(x-a)2+3!(x-a)3+

converge only when x = a. Others, like those in the important formulas

(12.412) e==1+x+2i--3 +4 +

(12.413)

(12.414)

cosx= l -2i+4i-66+.

6

sinx=x-33+5-7i+ ,