The Real And Complex Number Systems

sin 2s1 k sin 2 ksin k 2s−1
sin 2 k
s
∑ a j sin2j − 1k
j1
s
∑ a j sin 2 k sin2j − 1k
j1
s
∑ a j
j1
1 2
1 2
s
∑
j1
1 − cos2k
2
sin2j − 1k
by induction hypothesis
s
a j sin2j − 1k − ∑ a j cos2k sin2j − 1k
j1
s
∑ a j sin2j − 1k − 1 2 ∑ s
j1
j1
a j sin2j 1k sin2j − 3k
which is a linear combination of sink,...,sin2s 1k. Hence, we have proved the claim
by Mathematical Induction.
Remark: By the same argument, the series
n
∑
k1
cos 2p−1 k
is also bounded, i.e., there exists a positive number Mp such that
n
(6) Define ∑ k1
n
∑
k1
|cos 2p−1 k| ≤ Mp.
sinkx
: F
k n x, then F n x is boundedly convergent on R.
Proof: SinceF n x is a periodic function with period 2, andF n x is an odd function.
So, it suffices to consider F n x is defined on 0, . In addition, F n 0 0 for all n.
Hence, the domain I that we consider is 0, . Note that sinkx x
cosktdt. So,
0
n
F n x ∑
k1
0
x
∑
k1
0
x
0
x
sin kx
k
n
cosktdt
sinn 1 t − sin 1 t
2 2
2sin 1 t dt
2
sinn 1 2 t
t
dt
0
x
1
2sin t 2
− 1 t
k
sin n 1 2 t dt − x 2
which implies that
0
n 1 2
x sin t
t
|F n x| ≤
0
n 1 2 x sin t
t
dt
0
x t − 2sin t 2
2t sin t 2
dt
0
x t − 2sin t 2
2t sin t 2
sin n 1 2 t dt − x 2
sin n 1 2 t dt 2 .