Calculus: Analytic Geometry and Calculus, with Vectors

620

converge for each x. The geometric series in the formula

(12.415) 1 = 1 +x+x2+x3+

1 - x

Series

is an example of a power series in (x - 0) which converges for at least

one x different from 0 and diverges for at least one x. With each power

series in (x - a) which converges for at least one x different from a

and diverges for at least one x, there is associated a positive number R,

called the radius of convergence of the power series, such that the series

converges absolutely for each x for which Ix - al < R and diverges for

each x for which Ix - al > R. As Figure 12.42 indicates, the interval

Ix - al < R is called the interval of convergence of the series.

Interval of convergence

a-R a a+R s

Figure 12.42

We now give, without proof, some very useful information about power

series. Those who are interested in proofs should study the theory of

functions of a complex variable. If the power series in (12.43) converges

when lx - al < r and if for each such x we let f(x) be the number to

which the series converges, then

(12.43) f(x) = co + ci(x - a) + C2(X - a)2 + ca(x - a)' + ..

The function f thus defined is continuous over the interval Ix - a! < r

and, moreover, has derivatives of all positive orders which "can be

obtained by termwise differentiation," that is, when Ix - a! < r,

(12.441) f'(x) = cl + 2C2(X - a) + 3c3(x - a)2 + 4c4(x - a)3 +.

(12.442) f"(x) = 1.2c2 + 2.3c3(x - a) + 3.4c4(x - a)2

+ 4.5c5(x - a)3 +

(12.443) f(a)(x) = 1.2.3ca + 2.3.4c4(x - a) + 3.4.5c5(x- a)2

+ 4.5.6c6(x - a)' +

(12.444) f(4) (X) = 1.2.3.4c4 + 2.3.4.5c5(x - a)

+ 3.4.5.6c6(x - a)2 +

and so on, so that for each n = 1, 2, 3,

(12.445) f(-)(x) = n!c +(n 1 1)!c.+, (x - a)

+(n + 2) cn+2(x - a)2 +

Termwise integration as well as termwise differentiation is permissible,