✔ 最佳答案
1.
[1 + (1 / 3)] + [2 + (1 / 15)] + [3 + (1 / 63)] + ... + [8 + 1 / 255]
= 1 + 2 + 3 + ... + 8 + [1 / (1 * 3) + 1 / (3 * 5) + 1 / (5 * 7) + ... + 1 / (15 * 17)]
= 8(8 + 1) / 2 + (1 / 2)(1 / 1 - 1 / 3) + (1 / 2)(1 / 3 - 1 / 5) + ... + (1 / 2)(1 / 15 - 1 / 17)
= 28 + (1 / 2)[1 / 1 - 1 / 3 + 1 / 3 - 1 / 5 + ... + 1 / 15 - 1 / 17]
= 28 + (1 / 2)[1 / 1 - 1 / 17]
= 28 + (1 / 2)(16 / 17)
= 28 + 8 / 17
2. 設 x = (1 / 2 + 1 / 3 + ... + 1 / 2005)
(1/2+1/3+...+1/2006) (1+1/2+1/3+...+1/200 5)-(1+1/2+1/3+...+1/ 2006)(1/2+1/3+...1/2 005)
= [x + (1 / 2006)][1 + x] - [1 + x + (1 / 2006)][x]
= x + (1 / 2006) + x^2 + (1 / 2006)x - [x + x^2 + (1 / 2006)x]
= x + (1 / 2006) + x^2 + (1 / 2006)x - x - x^2 - (1 / 2006)x
= 1 / 2006
