A Complex Infinite Product

Calculus Level 5

If $$\displaystyle z = \prod_{k=1}^{\infty} \left( 1 + \dfrac{i}{2k^2} \right)$$, then what is the value of $$(1-i)z + \text{Arg}(z)$$?

Give your answer to 3 decimal places.

Notation: $$\text{Arg}(z)$$ represents the principal value of the angle of $$z$$.

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