A Controlled Spiral

Calculus Level 2

Suppose a particle moves in a right-angled left spiral on an xyxy-grid. That is, it moves a distance D1(x)D_{1}(x) in a straight line, stops, makes a right-angled turn to it's "left", travels a distance D2(x)D_{2}(x) in a straight line, stops, makes a right angled turn to its "left", travels a distance D3(x)D_{3}(x) in a straight line and continues in this fashion forever.

If Dn(x)=xn1(n1)!D_{n}(x) = \dfrac{x^{n-1}}{(n-1)!} for n1,n \ge 1, and if x=2015,x = 2015, then find the magnitude of the straight line distance between the particle's starting and finishing points.

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