General form of $\sum\limits_{i=1}^{k-1}\sin\left(\frac{k\pi}{i}\right)$ and $\prod\limits_{i=2}^{k-1}\sin\left(\frac{k\pi}{i}\right)$

Question

I am trying to find the general term for the following series:

$$\displaystyle\sum_{i=1}^{k-1}\sin\left(\frac{k\pi}{i}\right)$$

and

$$\displaystyle\prod_{i=2}^{k-1}\sin\left(\frac{k\pi}{i}\right).$$

I tried using the complex form of the sine function, but I failed to simplify it. Reviewing many posts I found examples of form $\sin(i\pi)$ but not of form $\sin\left(\frac{\pi}{i}\right)$

I don't have much experience in the area, but a good explanation would help me a lot. I appreciate your help.

Thanks in advance.

Answer 1

Concerning the summation$$S_k=\sum _{i=1}^{k-1} \sin \left(\frac{\pi k}{i}\right)$$ it is interesting to compare it to $$I_k=\int _{1}^{k} \sin \left(\frac{\pi k}{i}\right)\,di=\pi\, k\, (\text{Ci}(k \pi )-\text{Ci}(\pi ))$$ where appear the cosine integral function.

The asymptotics $$I_k =-\pi \, \text{Ci}(\pi )\, k+(-1)^{k+1} \left(\frac{1}{\pi k}-\frac{6}{\pi ^3 k^3}+\cdots\right)$$ and $-\pi \, \text{Ci}(\pi )\approx -0.231435$ which is almost identical to the result of the regression in my "non-answer".

Answer 2

This is not an answer.

If we consider $$S_k=\sum _{i=1}^{k-1} \sin \left(\frac{\pi k}{i}\right)$$ looking (from far away) at a plot of it for large values of $k$, it really looks like a straight line. A quick and dirty linear regression $(2 \leq k \leq 1000)$ with no intercept gives $(R^2=0.9995)$ $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & -0.231423 & 0.000163 & \{-0.231743,-0.231102\} \end{array}$$

Extrapolated to $k=10000$ this model would give $-2314.23$ while the exact value would be $-2305.89$ which is not so bad.

Concerning the product $$P_k=\prod _{i=2}^{k-1} \sin \left(\frac{\pi k}{i}\right)$$ for large values of $k$, its logarithm looks to be very linear with $k$.