I believe that the sum $$\lim\limits\_{m\to\infty}\sum\_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this number can be expressed without summation notation. If not, perhaps there is a good approximation?