# Step on it

Calculus Level 4

Suppose a particle, starting at the origin of a standard $$xy$$-grid, moves in a step-like manner, first going in a straight line right, (i.e., in the positive $$x$$-direction), then up, (i.e., in the positive $$y$$-direction), then right, up, and so on ad infinitum, such that the $$n$$th move has length $$d_{n} = \dfrac{(\ln(2))^{n-1}}{(n - 1)!}.$$

If the (magnitude of the) displacement between the starting and finishing points of the particle is $$\dfrac{\sqrt{a}}{b},$$ where $$a$$ and $$b$$ are positive integers with $$a$$ square-free, then find $$a + b.$$

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