Suppose that there is a positive random variable $X$. Also, assume that $E[X]<\infty$. Does that imply $X < \infty$ almost surely?
The last answer on a forum seem to suggest that it does. However, what if $$X=N\quad \text{w.p. }\frac{1}{N^3}$$ for $N\in {\mathbb N}$. Isn't the implication false?
Hint: $\{X < \infty\} = \bigcup_{n=1}^\infty\{X = n\}$ in your case. Also note that since $\sum \frac{1}{N^3}$ is not $1$, you need to scale your probabilities accordingly.