Un sir mai dificil

[Marius Dragoi]

Pentru orice numar natural \( n \geq 2 \) notam \( a_n = 2+ {\frac {1}{\sqrt 2}} + {\frac {1}{\sqrt 3}} +...+ {\frac {1}{\sqrt n}} \). Demonstrati ca \( 1+ {\frac{a_2}{2}} +{\frac {a_3}{3}} +...+ {\frac {a_n}{n} < a_{n-1} \).

Shortlist ONM 2004

[Marius Mainea]

Se arata prin inductie dupa n ca \( 1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n}>a_{n-1} \)