set $S_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i}$. Determine $$ \frac{E\big(\sum_{i=1}^{n}(X_{i}-S_{n})^{2}\big)}{Var(S_{n})}. $$ and I used law of large numbers to approach it
Since $S_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i}$ is an i.i.d sequence, from strong law of large numbers, we have $S_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i}\to E(X_1)$. And with the properties of expectations, \begin{align*} &\frac{E\big(\sum_{i=1}^{n}(X_{i}-S_{n})^{2}\big)}{Var(S_{n})} \\ =&\frac{E\big(\sum_{i=1}^{n}(X_{i}-S_{n})^{2}\big)}{E(S^2_n)-E(S_n)^2} \\ =&\frac{E\big(\sum_{i=1}^{n}(X_{i}-E(X_1))^{2}\big)}{E(X_1^2)-E(X_1)^{2}} \end{align*} what further steps should I take?