EDB — 210

[210] Note:exercise 2, written exam 15 January 2014. Let \((a_ n)_{ n ≥ 0}\) be a sequence of positive real numbers. Having defined \(s_ n =∑_{i=0}^ n a_ i \) prove that:

• the series \(∑_{n=0}^∞ a_ n\) converges if and only if the series \(∑_{n=0}^∞ a_{n}/s_ n\) converges;

• the series \(∑_{n=0}^∞ a_ n / (s_ n)^ 2\) converges.