Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 621

that is,

(12.45)

f." a

x

cl(t - a) dt + J c_(t -a)2dt

f.

z

co dt -}

a a

+ J c3(t - a)3 dt +.

a

= co(x - a) +- c1(x

2

a) + c(x

3

a)

4

+ C3 x

4

a +-

When Ix - al < r. Moreover, we can multiply (12.43) by g(x) and,

provided g is Riemann integrable over the interval froma to x, integrate

termwise to obtain

(12.46) f axf(t)g(t) dt=fax cog(t) dt +

fax

ci(t - a)g(t) dt

+ fax c2(t - a)2g(t) dl +

Putting x = a in (12.43) and the formulas that follow it gives the

remarkable formulas

(12.47) f(a) = co, f'(a) = c1, f"(a) = 2!C3, f(3)(a) = 3!c3,

fca>(a) = 4!c4, ...

Solving these equations for Co, c1, c2, and putting the results in

(12.43) gives the more remarkable formula

(12.48) f(x) = f(a) + f-a) (x- a)+t a) (x- a)2

+-----(x-a)3+. ...

These formulas show one of the ways in which the coefficients co, c3,
in a convergent power series can be determined in terms of the function
to which the series converges. The series in (12.48) is the Taylor series,
or Taylor expansion, off in powers of (x - a), and our work shows that
each convergent power series is the Taylor series of the function to which it
converges. In case a = 0, the Taylor series is sometimes called a Mac-
laurin series, but the practice has little justification and is being slowly
abandoned.
The following uniqueness theorem is used very often.
Theorem 12.481 If r > 0 and if the two power series Ebx(x - a)k and
ECk(x - a)k both converge to the same f(x) when Ix - al < r, so that

(12.482) f(x) = ho + bi(x - a) + b2(x - a)2 + Ox - a' < r)

(12.483) f(x) = co + c3(x - a) + c2(x - a)2 + Ox - al < r),

f (t) dt =

then bk = ck for each k.