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Infinite sum of non negative integers

  1. Mar 7, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider a sequence of non negative integers x1,x2,x3,...xn
    which of the following cannot be true ?
    ##A)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty##

    ##B)\sum ^{\infty }_{n=1} x_{n}= \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}< \infty##

    ##C)\sum ^{\infty }_{n=1} x_{n}< \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}< \infty##

    ##D)\sum ^{\infty }_{n=1} x_{n} \leq 5 \space and \space \sum ^{\infty }_{n=1} x_{n}^{2} \geq 25##

    ##E)\sum ^{\infty }_{n=1} x_{n}< \infty \space and \space \sum ^{\infty }_{n=1} x_{n}^{2}= \infty##

    2. Relevant equations


    3. The attempt at a solution

    A) is true when xn = n
    B)
    C) is true when xn = 1/n
    D)
    E)

    i cant find any ways to eliminate or finalise B,C, or D
     
  2. jcsd
  3. Mar 7, 2017 #2

    BvU

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    Hi,
    Let's work our way down the list. You did A already. Then:
    If ##x\ge1##, what do you know about ##x^2## in relation to ##x## ?
     
  4. Mar 7, 2017 #3

    PeroK

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    It says the sequences are integers. ##1/n## is not an integer.
     
  5. Mar 7, 2017 #4

    BvU

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    Hadn't even considered that ! was too focused on the fact that ##
    C)\sum ^{\infty }_{n=1} x_{n}\nless \infty## for 1/n
     
  6. Mar 7, 2017 #5
    ##x^2 \geq x## and the equality is only when x=1
    So in no case the sum of ##x_{n}## can exceed ##x_{n}^{2}##
    So B cannot be true .

    Am i correct Sir ?

    I never noticed that SIr !! thanks for pointing out ...

    if both the sequence contains only zeroes this is true ...
    So C is also eliminated.

    For D,

    Lets have first sequence : 5,0,0,0,0,0,.....
    So second sequence : 25,0,0,0,0,0,......

    So it is possible .

    for E,
    Only case where the first sum is less than infinity is finite number of positive terms. In that case second sum will also be finite.
    So E is also true

    So final answers B and E ?
    Am i correct now ???
    Thanks a lot both of you :)
     
  7. Mar 7, 2017 #6

    PeroK

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    Looks like you've got it.
     
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