A = 1 - 1/2 - 1/2^2 - 1/2^3 - 1/2^4 - 1/2^10
= 1 - (1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^10)
Đặt B = 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^10 ⇔ A = 1 - B
⇔ 2B = 1 + 1/2^2 + 1/2^3 + ... + 1/2^9
⇔ 2B - B = (1+ 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^9) - (1/2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^10)
⇔ B = 1 - 1/2^10
⇔ A = 1 - ( 1 - 1/2^10)
⇔ A = 1 - 1 + 1/2 ^10 = 1/2^10
***
B = 1 - 1/3 - 1/3^2 - 1/3 ^3 - 1/3^4 - ... - 1/3^10
= 1 - (1/3 + 1/3^2 + 1/3 ^3 + 1/3^4 + ... + 1/3^10)
Đặt A = 1/3 + 1/3^2 + 1/3 ^3 + 1/3^4 + ... + 1/3^10 ⇔ B = 1 - A
⇔ 2A = 1 + 1/3^2 + 1/3^3 + ... + 1/3^9
⇔ 2A - A = (1 + 1/3^2 + 1/3^3 + ... + 1/3^9) - (1/3 + 1/3^2 + 1/3 ^3 + 1/3^4 + ... + 1/3^10)
⇔ A = 1 - 1/3^10
⇔ B = 1 - ( 1 - 1/3^10)
⇔ B = 1 - 1 + 1/3^10 = 1/3^10
