An Odd Problem

Let $x_1, x_2, \dots x_{2k+1}$ be $2k + 1$ independent variables which are randomly chosen with uniform distribution in $(0, 1)$ , where $k$ is a non-negative integer.

$N = \sum_{i=1}^{2k+1}\left \lfloor{\frac{1}{x_i}} \right \rfloor$

Let $P(k)$ denote the probability that $N$ is odd.

Evaluate the limit below and enter your answer to three decimal places.

$\lim_{n\to \infty} \sum_{k=0}^{n} (2P(k)-1)$