-- nonconvergence of a series. with application in statistics

I am trying to show that
when Xi=+/- 3^(i-1) with probability 1/2 each, the mean Xbar goes
to +infty with probability 1/2 and goes to -infty with probability
1/2.

I have a hint telling me to evaluate the smallest value that (X1 + · ·
· + Xn) /n
can take on when Xn = 3^(n-1).

I suppose that the smallest value of Xbar_n when Xn = 3^(n-1)
must be reached when
X_{n-1} = - 3^(n-2)
X_{n-2} = + 3^(n-3)
X_{n-3} = - 3^(n-4)
....
X_{1} = + 3^(0)

In which case
(X1 + · · · + Xn) /n
=[3^(0) - 3^(1) + 3^(2)+......+3^(n-1)]/n
=[(1- (-3)^{n+1} )/(1+3)]/n
=[(1- (-3)^n)]/4n
Now (-3)^n/4n goes to -infty or +infty depending on n because
4n=o((-3)^n)

My argument is really not rigorous and is probably inaccurate, and I
still need
to show that -infty or +infty are reached with probability 1/2, and
1/2.
Can somebody help me improve my argument?

I appreciate your help/pointers.
.

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