@mathworker21 But it's an interesting point. This series is a blend between equations $(7)$ and $(3)$ in the link I provided. When $N = 0$ we have equation $(7)$ and as $N \to \infty$ we have equation $(3)$.

Yes, and that's a good point. If we let $N \to \infty$ in the formula in my question, we do get equation $(3)$. However there seems to be a 'sweet-spot' for the value of $N$. If $N$ is too small or too large the sum converges slower (implementation dependent values of $N$ course). For my implementation I used $N = 16$.