I am having a tough time deciding which Convergence test should I use to determine if the expression converges or diverges?
the expression is:
$$\lim_{n\to\infty}\frac{1}{n}\left(\sin\frac{\pi}{n}+\sin\frac{2\pi}{n}+\cdots+\sin{n\pi}{n}\right)$$
I tried coding it in Mathematica and it seems like the series converges, but I cannot decide which Convergence test should I use to determine that? Any help is appreciated.
Best Answer
Hint. If $\sin (x/2)\ne 0$ then $$\sum_{j=1}^n\sin jx=(\sin (x/2))^{-1}\sum_{j=1}^n(\sin jx)(\sin(x/2))=$$ $$=(\sin (x/2))^{-1}\sum_{j=1}^n(1/2)(\;\cos(jx-x/2)-\cos (jx+x/2)\;)=$$ $$=(\sin (x/2))^{-1}(1/2)(\;\cos (x/2)-\cos (nx+x/2)\,).$$ An example of a telescoping series, i.e. $\sum_{j=1}^n(\;F(j)-F(j+1)\;)=F(1)-F(n+1).$
We can also write $\sum_{j=1}^n\sin jx=$ $(2i)^{-1}\sum_{j=1}^n\exp (ijx)-(2i)^{-1}\sum_{j=1}^n\exp (-ijx)$ (where $i^2=-1$) and sum these two geometric series, and after a little re-arrangement, get the result above.