Trevor rides Jake piggyback I

Calculus Level 4

It's a beautiful Saturday morning. The wind is light yet refreshing, the sun warm and mellow, and the Trevor is riding Jake piggyback upon the verdant turf of the first golf course.

Starting at \((0,0)\), Trevor tells his serf to advance north by \(1\) meter and turn \(90^{\circ}\) clockwise. Jake, with his incredibly exact eyesight, does just that. At each nth step, he walks \(\frac{1}{n}\) meter and turns \(90^{\circ}\) clockwise.

What is the displacement in meters Trevor and Jake have travelled from the origin? If it can be expressed in the form \(\frac{1}{2} \sqrt{\ln^{2} a + \frac{\pi^{2}}{b}}\), where \(a\) and \(b\) are positive integers, what is \(a+b\)?

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