Condition for finite series: sum of squares finite + ?

[mercedesbenz]

Let u_n be a sequence of positive real number.
If \sum_{n=1}^{\infty}u_n^{2} finite + (condition??) then \sum_{n=1}^{\infty}u_n finite.
I want to find the condition.Please help me.

 

[HallsofIvy]

Any obvious condition would be that (u_{n+1}/u_n)^2 not go to 1 as n goes to infinity. The only way \sum_{n=1}^{\infty}u_n^{2} can converge is if lim (u_{n+1}/u_n)^2\le 1. If lim (u_{n+1}/u_n)^2< 1 then lim u_{n+1}/u_n< 1 also and so \sum_{n=1}^{\infty}u_n converges. Of course, that is a sufficient condition, not a necessary condition. It is still possible that a sequence for which lim u_{n+1}/u_n\le 1 will converge.

 

[mercedesbenz]

Thank you so much,HallsofIvy. In my first post. you know, this is my ploblem which I've tried to do it for 1 month. Thank you again.