文档介绍：Chapter Eleven
Taylor Series
Power Series
Now that we are knowledgeable about series, we can return to the problem of
investigating the approximation of functions by Taylor polynomials of higher and higher
degree. We begin with the idea of a so-called power series. A power series is a series of
the form
n
æç kö
å ck (x - a) .
è k= 0 ø
A power series is thus a sequence of special polynomials: each term is obtained from
the previous one by adding a constant multiple of the next higher power of (x - a).
Clearly the question of convergence will depend on x , as will the limit where there is one.
th k
The k term of the series is ck (x - a) so the Ratio Test calculation looks like
c (x - a)k +1 c
r(x) = lim k +1 = x - a lim k +1 .
k ®¥ k k®¥
ck (x - a) ck
Recall that our series converges for r(x) < 1 and diverges for r(x) > 1. Thus this
c
series converges absolutely for all values of x if the number lim k +1 = 0 . Otherwise, we
k ®¥
ck
c
have absolute convergence for | x - a | < lim k and divergence for
k®¥
ck +1
c c
| x - a | > lim k . The number R = lim k is called the radius of convergence,
k®¥ k®¥
ck +1 ck +1
and the interval | x - a |< R is called the interval of convergence. There are thus exactly
n
æç kö
three possibilities for the convergence of our power series å ck (x - a) :
è k= 0 ø
(i)The series conv