Say $x_i, i=1,2,3,\dots$ are i.i.d random variable. Let $S_n=\sum_{i=1}^nx_i$. We know from the law of large numbers that \begin{align} \frac{S_n}{n}\rightarrow E[x_1] \end{align} We also know if $var(x_1)<\infty$ that the convergence rate is $1/\sqrt{n}$.
Is it possible to say something about the convergence of $S_n$? Suppose $x_1$ has all moments finite. Can we say that $S_n\rightarrow nE[x_1]$? If not, why? Also it does not converge, can we add some more condition on the distribution of $x_1$ to make $S_n$ converge?