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.