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Infinite Sum

  1. Oct 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Is the proposition

    [itex]\sum^{\infty}_{n=1}[/itex]|x[itex]_{n}[/itex]|=0 ⇔[itex]\forall[/itex]n[itex]\in[/itex]IN x[itex]_{n}[/itex]=0

    true? If it is true how can we prove that ?
    2. Relevant equations



    3. The attempt at a solution
    I proved the [itex]\Leftarrow[/itex] side of proposition but i could not prove the [itex]\Rightarrow[/itex] side of proposition.
     
  2. jcsd
  3. Oct 5, 2012 #2
    Is it not true that [itex] \sum_n |x_n| > x_m [/itex] for all m?
     
  4. Oct 5, 2012 #3

    Mark44

    Staff: Mentor

    Is this what you're trying to prove?
    $$ \sum^{\infty}_{n=1}|x_{n}|= 0 \iff \forall n \in Z,~ x_{n} = 0$$

    I wasn't sure what you meant by IN. Also, one symbol you used (⇔) renders as a box in my browser.
    Edit: Now it's showing up. That's odd, it didn't before.

    Tip: Use one pair of tex or itex tags for the whole expression, rather than a whole bunch of them.
     
    Last edited: Oct 5, 2012
  5. Oct 5, 2012 #4
    Mark44

    The proposition which i want to prove is exactly the proposition which you write. I meant natural numbers by IN. Thanks for your suggestions.

    I am waiting for your answers.
     
  6. Oct 5, 2012 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Try a proof by contradiction. Assume one of the a's isn't zero.
     
  7. Oct 11, 2012 #6
    [itex]\sum\limits_{n=1}^{\infty}|x_{n}|=0\Rightarrow\lim \limits_{n\rightarrow\infty}\sum\limits_{k=1}^{n}|x_{k}|=0[/itex]. Since [itex]s_{n}= \sum\limits_{k=1}^{n}|x_{k}| [/itex] is an increasing sequence it can converge only its supremum. So [itex] \sup\limits_{n}s_{n}=0 [/itex]. Thus [itex] \forall n,\, 0\leq s_{n} = \sum\limits_{k=1}^{n}|x_{k}|\leq 0 [/itex] which means [itex] \forall n,\, x_{n}=0 [/itex].

    I made this proof. What do you think is there any mistake in this proof ?
     
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