# Stairway to heaven

Calculus Level 5

Let $$P_{n}$$ for any positive integer $$n$$ describe a particle's "stairway" path as follows:

A particle, starting at the origin of a standard $$xy$$-grid, first moves east, (i.e., in the positive $$x$$-direction), then up, (i.e., in the positive $$y$$-direction), then east, up, and so on for a total of $$n$$ moves, such that the $$k$$-th move has length $$\displaystyle \frac{1}{\sqrt{kn}}$$ for $$1 \le k \le n$$.

Now let $$D_{n}$$ be the displacement between the starting and finishing points, and let $$|P_{n}|$$ be the total distance traveled.

If $$\displaystyle S = \lim_{n \rightarrow \infty} (|P_{n}| - D_{n})$$, then find $$\lfloor 1000S \rfloor$$.

×