Two series of events

Topic: Probability spaces and events · more questions on this topic
‹ The algebra generated by two sets Continuity from above ›

Let A_n be independent with \mathrm{P}(A_n)=1/n for n\ge 1. What does Borel–Cantelli say about infinitely many occurrences? What changes if \mathrm{P}(A_n)=1/n^2?

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The harmonic series \sum 1/n diverges, and the events are independent, so the second Borel–Cantelli lemma gives \mathrm{P}(A_n\text{ infinitely often})=1. The series \sum 1/n^2 converges, so the first lemma gives probability 0 of infinitely many occurrences. The first lemma does not use independence.