Jake rides Trevor Piggyback IV

Without loss of generality we can assume that Trevor first heads east, (i.e., in the positive $x$ -direction), then north, (i.e., in the positive $y$ -direction), then west, south, east, north, and so on ad infinitum.

The net distance traveled in the positive $x$ -direction is then

$X = \displaystyle\sum_{k=0}^{\infty} (-1)^{k} \dfrac{1}{(2k + 1)^{2}} = K,$ Catalan's constant , which has a value of

$0.915965594177....$ (Interestingly, it is not known if this number is irrational.)

The net distance traveled in the positive $y$ -direction is

$Y = \displaystyle\sum_{k=1}^{\infty} (-1)^{k+1} \dfrac{1}{(2k)^{2}} = \dfrac{1}{4} \sum_{k=1}^{\infty} (-1)^{k+1} \dfrac{1}{k^{2}} = \dfrac{A}{4}$

where $A = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^{2}} - 2*\sum_{k=1}^{\infty} \dfrac{1}{(2k)^{2}} = B - \dfrac{B}{2} = \dfrac{B}{2},$

where $B = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^{2}} = \dfrac{\pi^{2}}{6},$ known as the solution to Basel's problem .

Thus $Y = \dfrac{B}{8} = \dfrac{\pi^{2}}{48},$ and so the magnitude of Trevor's displacement is

$\sqrt{X^{2} + Y^{2}} = \sqrt{K^{2} + \left(\dfrac{\pi^{2}}{48}\right)^{2}} = \boxed{0.939}$ to 3 decimal places.

Python 2.7:

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# JAKE AND TREVOR'S PIGGYBACK SIMULATION v1.0
from math import sqrt
infinity = 100000
class Piggyback():
    def __init__(self):
        self.x = 0
        self.y = 0
        self.direction = "up"
        self.step = 1
        self.turn = {\
            "right":"down",
            "down":"left",
            "left":"up",
            "up":"right"}
    def dist_from_origin(self):
        return sqrt(self.x**2 + self.y**2)
    def walk(self):
        if self.direction == "up":
            self.y += 1.0/self.step**2
        elif self.direction == "right":
            self.x += 1.0/self.step**2
        elif self.direction == "down":
            self.y -= 1.0/self.step**2
        elif self.direction == "left":
            self.x -= 1.0/self.step**2
    def go(self):
        while self.step <= infinity:
            self.walk()
            self.step += 1
            self.direction = self.turn[self.direction]
jake_and_trevor = Piggyback()
jake_and_trevor.go()
print "Answer:", round(jake_and_trevor.dist_from_origin(),3)