Let \((\Omega,\mcf,P)\) be a probability space, let \((E_n)_{n\geq 1}\) be a sequence of events, and let \(\limsup_n E_n\) be its limit superior.
If \(\sum_{n\geq 1} P(E_n) < \infty\), then \(P(\limsup_n E_n) = 0\).
If the events \(E_n\) are independent and \(\sum_{n\geq 1} P(E_n) = \infty\), then \(P(\limsup_n E_n) = 1\).
For (1), \(\limsup_n E_n\subseteq\bigcup_{n\geq N}E_n\) for every \(N\), so \(P(\limsup_n E_n)\leq\sum_{n\geq N}P(E_n)\to 0\) as \(N\to\infty\).
For (2), the complement of \(\limsup_n E_n\) is \(\bigcup_{N\geq 1}\bigcap_{n\geq N}E_n^c\). For \(M\geq N\), independence and \(1-t\leq e^{-t}\) give \[ P\qty{\bigcap_{n=N}^{M}E_n^c} = \prod_{n=N}^{M}(1-P(E_n))\leq\exp\qty{-\sum_{n=N}^{M}P(E_n)}\to 0 \quad (M\to\infty), \] so \(P\qty{\bigcap_{n\geq N}E_n^c} = 0\) for every \(N\), and the countable union of these null events is null.