limit of a function tends to Dirac delta function

Question

Let

$f_K(x)=\frac{1}{\sqrt{K}}\sum_{k=0}^{K-1}\exp\{i\pi kx\}=\exp\{i\pi(K-1)x/2\}\frac{\sin(\pi Kx/2)}{\sqrt{K}\sin(\pi x/2)}$

where $i=\sqrt{-1}$.

I want to know whether the limit of $f_K(x)$ is a Dirac delta function when $K\to\infty$?

Answer 1

Well, $$ \int_{-\pi}^\pi f_K(x)\,dx=\frac{2\,\pi}{\sqrt K} $$ converges to $0$ as $K\to\infty$, so the answer is no.