The Real And Complex Number Systems

n
S n ∑−1 k1 1
3k − 2 − 1
3k − 1
k1
n
∑−1 k 1
k1
n
−∑
k1
−
−
−
3n
∑
k1
n
3k − 1 ∑
k1
−1 3k−1 1
−1 k1 1
3k − 2
n
3k − 1 − ∑−1 3k−2 1
3k − 2
k1
n
n
∑−1 3k−1 1
3k − 1 ∑−1 3k−2 1
3k − 2
k1
k1
3n
∑
k1
3n
∑
k1
−1 k
k
−1 k
k
−1 k1
k
n
− ∑
k1
− 1 3 ∑ k1
n
−1 3k
3k
n
−1 k
k
−1 k1
k
− 1 3 ∑ k1
→ 2 log 2.
3
So, the series has the sum 2 log 2.
3
Remark: There is a refernece around rearrangement of series. The reader can see the
book, An Introduction to Mathematical Analysis by Loo-Keng Hua, pp 323-325.
(Chinese Version)
8.19 Let c n a n ib n ,wherea n −1 n / n , b n 1/n 2 . Show that ∑ c n is
conditioinally convergent.
Proof: It is clear that ∑ c n converges. Consider
∑|c n| ∑ 1 n 2 1 n 4 ∑ 1 n 1 1 n 2 ≥ ∑ 1 n
Hence, ∑|c n| diverges. That is, ∑ c n is conditioinally convergent.
Remark: Wesay∑ c n converges if, and only if, the real part ∑ a n converges and the
imaginary part ∑ b n converges, where c n a n ib n .
8.20 Use Theorem 8.23 to derive the following formulas:
n
(a) ∑ k1
logk
1 k 2 log2 n A O logn
n
(A is constant)
Proof: Letfx logx
x define on 3, , then f ′ x 1−logx 0on3, . So, it is
x 2
clear that fx is a positive and continuous function on 3, , with
log x
lim x→
fx lim x→ x
So, by Theorem 8.23, wehave
lim x→
1 x 0byL-Hospital Rule.