The Real And Complex Number Systems

have
∫ x
0
∞∑
t 2n − 1 2
n=0
∞∑
t n =
n=0
=
∫ x
0
∫ x
0
1
1 − t − 1
2 2 (1 − t) dt
(
1 1
2 1 − t + 1 )
1 + t
= 1 log (1 + x) .
2
− 1 2
( ) 1
dt
1 − t
And as x = 1,
∞∑ (
x 2n+1 / (2n + 1) − x n+1 / (2n + 2) )
n=0
=
=
∞∑
n=0
∞∑
n=0
1
2n + 1 − 1
2n
(−1) n+1
n + 1
by Theorem8.14.
= log 2 by Abel’s Limit Theorem.
9.22 Prove that ∑ a n sin nx and ∑ b n cos nx are uniformly convergent on
R if ∑ |a n | converges.
Proof: It is trivial by Weierstrass M-test.
9.23 Let {a n } be a decreasing sequence of positive terms. Prove that
the series ∑ a n sin nx converges uniformly on R if, and only if, na n → 0 as
n → ∞.
Proof: (⇒) Suppose that the series ∑ a n sin nx converges uniformly on
R, then given ε > 0, there exists a positive integer N such that as n ≥ N,
we have
∣ ∣∣∣∣ 2n−1
∑
a k sin kx
< ε. (*)
∣
Choose x = 1
2n , then sin 1 2
k=n
≤ sin kx ≤ sin 1. Hence, as n ≥ N, we always
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