6. Given the set $T=\{1,2, \cdots, 2010\}$, for each non-empty subset of $T$, calculate the reciprocal of the product of all its elements. Then the sum of all such reciprocals is $\qquad$
6. 2010 .

Notice that, in the expansion of the product
$$
(1+1)\left(1+\frac{1}{2}\right) \cdots\left(1+\frac{1}{2010}\right)
$$

the $2^{2010}$ terms are all the reciprocals of positive integers, and each is precisely the reciprocal of the product of the numbers in one of the $2^{2010}$ subsets of set $T$. Removing the term 1, which corresponds to the empty set, the sum of the reciprocals is
$$
2 \times \frac{3}{2} \times \frac{4}{3} \times \cdots \times \frac{2011}{2010}-1=2010 .
$$