Invoking Complex Analysis!

$\large x_r = e^{\frac{i\pi}{n^r}}$

$\displaystyle\begin{aligned}\Rightarrow \prod\limits_{r = 1}^\infty x_r &= x_1x_2x_3x_4\cdots\infty\\ &=e^{\dfrac{i\pi}{n^1}}e^{\dfrac{i\pi}{n^2}}e^{\dfrac{i\pi}{n^3}} \cdots \infty\\ &=e^{i\pi\left(\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\cdots\infty\right)}\\ &=e^{i\pi\dfrac{1}{n}\times\dfrac{1}{1-\frac{1}{n}}}\\ &=e^{i\pi\times\dfrac{1}{n-1}}\\ &=\cos\dfrac{\pi}{n-1}+i\sin\dfrac{\pi}{n-1}\\\\ n = 5, \\ \prod \limits_{r=1}^{\infty} &= \cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\\ &=\dfrac{1}{\sqrt{2}}+\dfrac{i}{\sqrt{2}}\end{aligned}$

$\Large\therefore \boxed{Re\left(\prod\limits_{r=1}^{\infty} x_r\right) + Im\left(\prod\limits_{r=1}^{\infty} x_r\right) + \left|\prod\limits_{r=1}^{\infty} x_r\right| =\huge\sqrt{2}+1}\Large\approx 2.4142$