Answer: Assume as known the (true) fact that the alternating harmonic series (1) is

Chapter 9, Problem 82

(choose chapter or problem)

Assume as known the (true) fact that the alternating harmonic series

(1) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\cdots\)

is convergent, and denote its sum by Rearrange the series (1) as follows:

(2) \(1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+\cdot \cdot .\)

Assume as known the (true) fact that the series (2) is also convergent, and denote its sum by S. Denote by \(s_{k}, S_{k}\) the kth partial sum of the series (1) and (2), respectively. Prove

the following statements.

(i) \(S_{3 n}=S_{4 n}+\frac{1}{2} S_{2 n}\)

(ii) \(S \neq S\)

Text Transcription:

1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+...

1+1/3-½+1/5+1/7-1/4+1/9+1/11-1/6+...

s_k,S_k

S_3n=s_4n+1/2s_2n

S neq s