Ta có
$A = \left( \dfrac{1}{10} - 1 \right) \left( \dfrac{1}{11} - 1 \right) \left( \dfrac{1}{12} - 1 \right).\dots.\left( \dfrac{1}{100} - 1 \right)$
$= -\dfrac{9}{10} . \left( -\dfrac{10}{11} \right) . \left( -\dfrac{11}{12} \right) .\dots.\left( -\dfrac{99}{100} \right)$
Số dấu trừ là
$(100 - 10) + 1 = 91$
Vậy ta có
$A = (-1)^{91} . \dfrac{9}{10} . \dfrac{10}{11} . \dfrac{11}{12} . \dots . \dfrac{99}{100}$
$= - \dfrac{9}{100}$
Vậy $A = -\dfrac{9}{100}$.
