Finding a closed form expression for $S=\sum_{i=2}^{n-1}\csc({\frac{\pi}{i}})$ or approximation?

Question

Consider $S=\sum_{i=2}^{n-1}\csc({\frac{\pi}{i}})$

How can we find a general formula for S using trigonometry identities or complex numbers ?

If a closed from cannot be found then how can we approximate it?

This is quite similar Finding a closed form expression for $\sum_{i=1}^{n-1}\csc{\frac{i\pi}{n}}$

Answer 1

Let $\, S_n := \sum_{i=2}^{n-1} \csc(\pi/i). \,$ Note the Laurent series expansion in powers of $\,1/i\,$: $$ \csc\left(\frac{\pi}i\right) = \frac{i}{\pi} + \frac{\pi}{6i} + \frac{7\pi^3}{360i^3} + O\left(\frac1{i^5}\right). $$ By summing we get $$ S_n \approx (n^2-2)/(2\pi) - C + H_{n-1}\pi/6 \quad \text{ where } \quad C \approx 0.7.$$ We can get closer approximations by using generalized Harmonic numbers.