The Real And Complex Number Systems

n
u n ∑−1 k a k
k1
n
∑−1 k s k − s k−1
k1
n
∑−1 k s k ∑−1 k1 s k−1
k1
n
∑
k1
n
k1
n
−1 k s k ∑
k1
−1 k s k −1 n1 s n
which implies that
2v n −1 n1 s n
v n 1 2 u n −1 n s n /2.
(b) ∑
n1
−1 n s n is C,1 summable and has Ces’aro sum 1 ∑ −1 n a
2 n1 n .
Proof: First, lim n→ u n exists since it is an alternating series. In addition, since
lim n→ a n 0, we know that lim n→ s n /n 0byTheorem 8.48. Hence,
Consider by (a),
v n
n
∑ k1
v k
n
u n
2n −1n s n
2n
1 n
2
∑ k1
∑ n
u
k1 k
2n
→ 1 2 lim n→ u k
→ 0asn → .
u k 1 n
2
∑ k1
−1 k s k
n
v n
2n
1 2 ∑ −1 n a n
n1
by Theorem 8.48.
(c) ∑ n1
−1 n 1 1 2 ... 1 n − log 2 C,1.
Proof: By (b) and ∑ n1
Infinite products
−1 n
n
− log 2, it is clear.
8.39 Determine whether or not the following infinite products converges. Find the
value of each convergent product.
(a) n2
1 −
2
nn1
Proof: Consider
we have
1 − 2
nn 1
n − 1n 2
nn 1
,