Let $\left\{Y_n\right\}_{n=1}^{\infty}$ be a sequence of random variables, $Y_n \text{ i.i.d. } Z\equiv N(0,1)$. I want to find the limit of: $$K_n=\sqrt{n}\:\dfrac{Y_1+\cdots+Y_n}{{Y_1}^2+\cdots+{Y_n}^2}$$
I tried using $\frac{1}{n}\left(Y_1+\cdots+Y_n\right)\longrightarrow 0$ and $\frac{1}{n}\left({Y_1}^2+\cdots+{Y_n}^2\right)\longrightarrow 1$, and then using that $K_n$ is similar to the definition of Student's T but then I didn't get anywhere.