EDB — 0F4

Exercises

1. [0F4]Note:Written exam of 4th Apr 2009, exee 1.(Proposed on 2022-12-13) Given a sequence \((a_ n)_{n}\) of strictly positive numbers, it is said that the infinite product \(∏_{n=0}^∞ a_ n\) converges if there exists finite and strictly positive the limit of partial products, i.e.

   \[ \lim _{N→+∞}∏_{n=0}^ Na_ n ∈ (0,+∞)\quad . \]

   Prove that

   1. if \(∏_{n=0}^∞ a_ n\) converges then \(\lim _{n→+∞}a_ n=1\);

   2. if the series \(∑_{n=0}^∞|a_ n-1|\) converges, then it also converges \(∏_{n=0}^∞ a_ n\);

   3. find an example where the series \(∑_{n=0}^∞(a_ n-1)\) converges but \(∏_{n=0}^∞ a_ n=0\).