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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