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Sum math problem

  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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    Are you looking for a necessary or a sufficient condition?
  4. Jan 28, 2008 #3
    and real?
  5. Jan 28, 2008 #4


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    IIRC, then there is a theorem like this:

    Given the sequence of positive real number (un)

    The series [tex]\sum_{n = 1} ^ {\infty} u_n[/tex] converge, if and only if [tex]\lim_{n \rightarrow \infty}(u_n \times n ) = 0[/tex].

    Let's see if you can prove this theorem. :)

    Now, using the above theorem, can you try to work out the problem? :)
  6. Jan 28, 2008 #5
    thank you so much for your advice,VietDao29.I will try to do it again.
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