# A random variable's expectation goes to infinity but its value converges to zero

I am currently working on a problem associated with the random variable $Z_n$ such that $Z_n = Z_{n-1}X_n$, where $X_i's$ are i.i.d. random variable with the calculated expectation to be greater than 1.

So after calculation I found that $E[Z_n] \to \infty$ as $n \to \infty$, how ever $Z_n$ converges to 0 almost surely after I taking log and apply the SLLN. / I am confused on why the results are different but still compatible? Thanks!

$L_1$ convergence (which means $E(|X_n-X|)\to 0)$and almost sure convergence are incomparable conditions where neither implies the other. Yours must be an example where almost sure convergence doesn't imply $L_1$ convergence. There are simpler examples, like
$$X_n = \left\{\begin{array}{ll}0&\mbox{with probability 1-1/n^2}\\n^3&\mbox{with probability \frac{1}{n^2}}\end{array}\right.$$
There can also be cases where the sequence strays infinitely often so as not to converge but nonetheless doesn't stray very far so doesn't impact the mean. For instance $$X_n = \left\{\begin{array}{ll}0&\mbox{with probability 1-1/n}\\1&\mbox{with probability \frac{1}{n}}\end{array}\right.$$ almost surely does not converge, but $E(|X_n|)\to 0.$