Invoking Complex Analysis!

Algebra Level 4

If \(\large x_r = \cos \left( \dfrac{\pi}{5^r} \right) + i \sin \left( \dfrac{\pi}{5^r} \right) \)

Evaluate the value of:

\[\large Re\left(\prod_{r=1}^\infty x_r \right) + Im\left(\prod_{r=1}^\infty x_r \right) + \left|\prod_{r=1}^\infty x_r \right| \]

where \(Re(z), \ Im(z)\)and \( |z|\) denotes the real part, the imaginary part and the magnitude of \(z\) respectively.

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