Let $S$ be the sum. Then
$$1005 - S = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \dots + \frac{1}{2011} = T.$$
Then $S$ is an integer if and only if $T$ is an integer. Let $M = 3 \cdot 5 \cdot 7 \cdot 9 \dots 2009$. If $T$ is an integer then $MT$ is an integer.
$$MT = \frac{M}{3} + \frac{M}{5} + \frac{M}{7} + \frac{M}{9} + \dots + \frac{M}{2009} + \frac{M}{2011}.$$
Each term is an integer except the last one which is not since $2011$ is prime. Hence $T$ is not an integer.