A Controlled Spiral

Calculus Level 3

Suppose a particle moves in a right-angled left spiral on an \(xy\)-grid. That is, it moves a distance \(D_{1}(x)\) in a straight line, stops, makes a right-angled turn to it's "left", travels a distance \(D_{2}(x)\) in a straight line, stops, makes a right angled turn to its "left", travels a distance \(D_{3}(x)\) in a straight line and continues in this fashion forever.

If \(D_{n}(x) = \dfrac{x^{n-1}}{(n-1)!}\) for \(n \ge 1,\) and if \(x = 2015,\) then find the magnitude of the straight line distance between the particle's starting and finishing points.

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