The Real And Complex Number Systems

which implies that
which implies that (if |z| 1)
So,
n
k0
n
1 − z 1 z 2k
1 − z 2n1
k0
1 z 2k
1 − z2n1
1 − z
→ 1
1 − z
1 z 2n
1
1 − z .
n0
as n → .
8.40 If each partial sum s n of the convergent series ∑ a n is not zero and if the sum
itself is not zero, show that the infinite product a 1 n2
1 − a n /s n−1 converges and has the
value ∑ n1
a n .
Proof: Consider
n
n
s
a 1 1 a k /s k−1 a 1 k−1 a k
s k−1
k2
k2
n
a 1
k2
s k
s k−1
s n → ∑ a n ≠ 0.
So, the infinite product a 1
n2
1 − a n /s n−1 converges and has the value ∑ n1
a n .
8.41 Find the values of the following products by establishing the following identities
and summing the series:
(a) n2
1 − 1 2 ∑ 2
2 n −2 −n .
n1
Proof: Consider
we have
1 − 1
2 n − 2 2n − 1
2 n − 2 1 2 2n − 1
2 n−1 − 1 ,
n
k2
1 − 1
2 k − 2
n
k2
1
2 2k − 1
2 k−1 − 1
n
2 −n−1
k2
2 k − 1
2 k−1 − 1
2 −n−1 2 n − 1
2 −n−1 2 n−1 ...1
1 ... 1
n
∑
k1
n
2 ∑
k1
1
2 k−1
2 n−1
1
2 k .