I want to show that the law of large numbers holds/doesn't hold for the sequence of independent random variables $P\{X_k = \pm 2^k\} = \frac{1}{2}$
A sufficient condition is $\frac{s_n}{n} \rightarrow 0$ where $s_n$ is the sum of the variances of the variables. However, this is clearly not true so I'm trying to show that the law of large numbers fails to hold.
Assume they are independent, and $S_n=\sum\limits_{k=1}^n X_k$.
Actually for this simple example we can list all possibilities of the outcome: using induction or binary system, we know $S_n$ is symmetric with $2^{n-1}$ positive outcomes with equal probability $1/2^{n}$: $2,6,10,\cdots,2^{n+1}-2$. So there are at most $\lfloor (n+2)/4\rfloor<n$ number of the outcomes can be larger than $n$, which means $$P(|\frac{S_n}{n}|>1)\geq 1-\frac{n}{2^{n}}\rightarrow 1$$
if $n\rightarrow\infty$. So it cannot satisfy the weak law of large numbers.
If they are dependent, I wonder whether there is an example that WLLN holds--will edit later if I find one or disprove it.