Calculus: Analytic Geometry and Calculus, with Vectors

632 Series

when jxj < R. Persons willing to go fishing even when no fish are caught can
try to prove the result with rudimentary equipment. To simplify writing,
let a = 0. Then, when txol < R and Ix( < R,

n

f(x) Ckxk = Jim I Ckxk,

k=0 n-- k=0

m n

f(xo) = ckxo = lim ckxo.

k=0 n-"O k=0

so

f(x') - f(xo) _

n

lim ck(xx - xo)x

n-+GO k=1

and

n

f(x) - f(xo)= lim C c1

+

ck(xk-' + xk-2xo +... + x0

X - xo n-- k=2I

If we can prove that the limits exist, we can take limits as x approaches xo to

obtain

n

f'(xo) = lim lim 1c, + ck(xk-1 + xk-2xo +. .. + xo-1).

x-.xo n-- - k=2

If we can prove that the same result is obtained by interchanging the order in

which limits are taken, we obtain

n

f'(xo) = lim lim I C1 + I Ck(xk-' + xk-2xo + + 40-1)

n-+m x--.x0 k-2

and hence

m

f'(xo) = lim Icl + kckxo-11= kckxo-1.

n-- k=2 k=1

Our fishing expedition can be ended with the remark that we ran into questions

involving iterated limits and change of order of limits that swamped us.

12.5 Taylor formulas with remainders In Section 12.4, we started

with given convergent power series and found that these series are the

Taylor series of the functions to which they converge. In this section

we start with a given function f and study the general aspects and further

applications of a method we have previously employed in special cases

to obtain power series expansions of ex, cos x, and sin x. We suppose

that a and x are confined to an interval over which f has all of the con-

tinuous derivatives we want to use. Then

(12.51) f(x) = f(a) + fax f' (t) dt.