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Condition for finite series: sum of squares finite + ?

  1. Jan 28, 2008 #1
    Let [tex]u_n[/tex] be a sequence of positive real number.
    If [tex]\sum_{n=1}^{\infty}u_n^{2}[/tex] finite + (condition??) then [tex]\sum_{n=1}^{\infty}u_n[/tex] finite.
    I want to find the condition.Please help me.
  2. jcsd
  3. Jan 28, 2008 #2


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    Any obvious condition would be that [itex](u_{n+1}/u_n)^2[/itex] not go to 1 as n goes to infinity. The only way [tex]\sum_{n=1}^{\infty}u_n^{2}[/tex] can converge is if [itex]lim (u_{n+1}/u_n)^2\le 1[/itex]. If [itex]lim (u_{n+1}/u_n)^2< 1[/itex] then [itex]lim u_{n+1}/u_n< 1[/itex] also and so [tex]\sum_{n=1}^{\infty}u_n[/tex] converges. Of course, that is a sufficient condition, not a necessary condition. It is still possible that a sequence for which [itex]lim u_{n+1}/u_n\le 1[/itex] will converge.
  4. Jan 28, 2008 #3
    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.
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