General form of $\sum_{i=1}^{k} \frac{1}{k}\tan(\frac{i\theta}{k})$

Question

I have tried calculating this equation by setting a large value of $k$. given $\theta$, $$S = \sum_{i=1}^{k} \frac{1}{k}\tan\bigg(\frac{i\theta}{k}\bigg)$$ When increasing $k$, $S$ seems converging to one constant value. I wonder there is a general form of this series. If yes, how to determine the general form of this series?

I appreciate your help. Thank you very much

theta = 0.927295218001612
k=1000          S=0.551543821253878
k=10000         S=0.550943685250616
k=100000        S=0.550883683890583
k=1000000       S=0.550877683876961
k=10000000      S=0.550877083876862

Answer 1

As $k$ tends to $\infty$, the sum converges to the integral $$I=\int_0^1\tan(\theta x)dx=\frac{1}{\theta}\left[-\ln|\cos(\theta x)|\right]_0^1=-\frac{1}{\theta}\ln|\cos\theta|$$

Setting $\theta= 0.927295218001612$ leads to the result $0.550877017210179$, which is the value the sum will converge to