Evaluate the limit of $(1/n)( \sin (\phi/n) + \sin (2\phi/n) + … + \sin ((n-1)\phi/n))$

I tried to evaluate the limit of the following sequence:

$$ \frac {1}{n} \left( \sin \frac{\phi}{n} + \sin \frac{2\phi}{n} + \, … \, + \sin \frac{(n-1)\phi}{n} \right)$$

but i can’t seem to come up with anything. Asymptotic estimates don’t help, i tried substituting phi with a random x and the graph looks like a sinusoidal odd function with something like an exponential multiplied. So the limit should be finite. That’s about as far as i got.

Solution 1:

You could use the following $$ \sin(x)+\sin(2x)+\dots+\sin(kx)=\dfrac{\sin\left(\frac{k+1}{2}x\right)\cdot \sin\left(\frac{kx}{2}\right)}{\sin\left(\frac{x}{2}\right)} $$ Then we taking $x=\frac{\phi}{n}$ and $k=n-1$ we have \begin{align} \frac {1}{n} \left( \sin \frac{\phi}{n} + \sin \frac{2\phi}{n} + \dots + \sin \frac{(n-1)\phi}{n} \right)&=\frac{1}{n}\dfrac{\sin\left(\frac{n}{2}\frac{\phi}{n}\right)\cdot \sin\left(\frac{(n-1)}{2}\frac{\phi}{n}\right)}{\sin\left(\frac{\phi}{2n}\right)}\\ &=\frac{\sin(\frac{\phi}{2})\cdot \sin\Big((1-\frac{1}{n})\frac{\phi}{2}\Big)}{\frac{\sin(\phi/2n)}{1/n}} \end{align} Doing $n\to+\infty$ we take $$\displaystyle\lim_{n\to +\infty}\frac {1}{n} \left( \sin \frac{\phi}{n} + \sin \frac{2\phi}{n} + \dots + \sin \frac{(n-1)\phi}{n} \right)=\frac{2}{\phi}\sin^2(\phi/2).$$

You can also use Riemann sums, for example in this case \begin{align} \displaystyle\lim_{n\to +\infty}\frac {1}{n} \left( \sin \frac{\phi}{n} + \sin \frac{2\phi}{n} + \dots + \sin \frac{(n-1)\phi}{n} \right)&=\lim_{n\to +\infty}\sum_{k=1}^{n-1}\sin(\frac{k\phi}{n})\frac{1}{n}\\ &=\int_0^1 \sin(\phi x)dx\\ &=\frac{1}{\phi}(1-\cos(\phi))\\ &=\frac{2}{\phi}\sin^2(\phi/2) \end{align}