
“real: chapter_10” — 2011/5/22 — 23:26 — page 15 — #15
Power Series and Special Functions 10-15
We now choose r such that 0 < r < 1 and restrict x to |x|≤r.By
Theorem 10.2.3, we know that s
n
(x) → s(x) uniformly for |x|≤r as
n →∞. We now apply Corollary 9.3.27 and get
log(1 + x) =
x
0
dt
1 + t
=
x
0
∞
n=0
(−1)
n
t
n
=
∞
n=0
(−1)
n
x
n+1
(n + 1)
= x −
x
2
2
+
x
3
3
+···
valid for |x|≤r < 1. Since r can be chosen as close to 1 as we want,
the result is now valid for all |x| < 1.
The property that lim
x→∞
x
−n
x
n
e
−x
= 0 for all n can be converted as
the following property of log x. lim
x→∞
x
−n
log x = 0 for all n > 0 and
lim
x→0+
x
n
log x = 0 for all n > 0. Indeed, log x →−∞as x → 0+ and
log ...