>>10873880cos changes sign whenever n crosses a value of pi/(2*alpha) + (pi/alpha)*k, where k is an integer. In fact, the cumulative sum increases by 1 for each additional value crossed, so k and N_n(alpha) are the same, up to some added constant. We can compute the largest crossed k from n by rearranging the crossing formula, but we also need to add a correction term, so N_n(alpha) = (alpha/pi)*n - 0.5 + c - e(n), where c is the added constant, and e(n) is the correction term between 0 and 1. 0.5, c, and - e(n) are all bounded, so diving them by n will tend to 0 as n goes to infinity. So only the (alpha/pi)*n term matters, and, when divided by n, it tends to alpha/pi as n goes to infinity.
