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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“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
x0+
x
n
log x = 0 for all n > 0. Indeed, log x →−∞as x 0+ and
log ...
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Publisher Resources

ISBN: 9781299447561Publisher Website