Little Big Sum

Calculus Level 5

When

$S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|,$

the value of $$S$$ is of the form

$\dfrac{1}{2}\sqrt{\log^2(\tan x)+1},$

where $$0<x<\dfrac{\pi}{4}.$$ Find $$\lfloor 1000x \rfloor.$$

Details and Assumptions

• $$i$$ is the imaginary unit, and not a variable.

• $$\lfloor \cdots \rfloor$$ is the floor function, or greatest integer function.

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