Convergence and limit of $\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac jn\right)-\cos\left(\frac{j-1}n\right)\right\}$ The title says it all - I'm trying to find a way of proving the convergence and evaluating the limit of $a_n=\sum_{j=1}^n\sin\left(\frac{j-1}n\right)\left\{\cos\left(\frac jn\right)-\cos\left(\frac{j-1}n\right)\right\}$. This exercise is from the field of integration theory, so probably Riemann sums are to be used on this one but I can't find any way that makes sense. I'd appreciate any hints.

Rather than go through all the ugly trigonometric details of this particular problem, I would suggest reading about the Riemann-Stieltjes integral. We have that
$$\sum_{j=1}^nf(\tfrac{j-1}n)[g(\tfrac{j}n)-g(\tfrac{j-1}n)]\to\int_0^1f(x)\mathrm dg(x)$$
as $n\to\infty$. If $g$ is differentiable, $\mathrm dg(x)=g'(x)\mathrm dx$. Plugging in $f=\sin$, $g=\cos$ will give us the solution.
