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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