522 Chapter 8 • Infinite Series of Real Numbers
Using the Maclaurin series for ex and the algebra of power series (8.6.9),
we find that
which is valid for all x E R D
Example 8.7.7 Taylor series for sinx about c.
Using the Maclaurin series for sinx and cosx, and the algebra of power
series Theorem (8.6.9), we find that for all x E JR,
sin x = sin[c + (x - c)] =sin ccos(x - c) +cos csin(x - c)
. oo (-l)k(x- c)2k+l oo (-l)k(x-c)2k
= smc L (2k + 1)! + cosc L (2k)!
k=O k=O
=sinc[l-(x-c)2 + (x-c)4 - ···J+cosc[(x-c)- (x-c)3 + (x-c)5 -···]
2! 4! 3! 5!
L
oo ak(x - c)k h { (-l)k/^2 sin c if k is even }
= w ere ak = ·
kl ' (-l)(k-l)/^2 cosc if k is odd
k=O
DTHE BINOMIAL SERIESRecall the binomial theorem,^10 which says that "in E N, and "ix, y E JR,w h ere (n) k - k!(nn! -k)!. If we wnte · (n) k - n(n-l)(n-2) k! .. -(n-k+l) , t en h t h" is 1ormu c^1 a
makes sense even when k > n. In fact, for natural numbers k > n, (~) = 0.
Because of this we have the infinite series representation
00
(x +Yr= L G)xn-kyk.
k=O
As a young man, Newton developed a power series expansion of (1 + x)°'
when a is not a positive integer. He considered his derivation and analysis of
this series among his finest achievements. We now call this series the binomial
series, but we use more modern methods in its analysis.
Definition 8.7.8 Given an arbitrary real number a, the series
00
2: (~)xk,
k=O
- See Exercise 1.3.24.