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Divergence of absolute series

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that following statement is true:
    If Σa_n diverges, then Σ|a_n| diverges as well.

    2. Relevant equations

    Comparison Test:
    If 0 ≤ a_n ≤ b_n for all n ≥ 1, and if Σa_n diverges, then Σb_n diverges as well.

    3. The attempt at a solution

    I tried to prove the statement by using the Comparison Test with a_n = a_n and b_n = |a_n|, but the condition for the Comparison Test is that both sequences must be greater than or equal to zero, which is not true for this problem. I would like to know if using the Comparison Test is a right approach to prove this statement.

    Thank you in advance.
  2. jcsd
  3. Nov 13, 2008 #2
    Think of the Cauchy criterion; that is, [itex]\sum a_n[/itex] converges if and only if for all [itex]\epsilon >0[/itex] there exists an integer N such that [itex]\left| \sum_{k=n}^ma_k \right| \leq \epsilon[/itex] if [itex] m\geq n\geq N[/itex].

    Now think of what you have to prove and the fact that

    [tex]\left| \sum_{k=n}^ma_k \right| \leq \sum_{k=n}^m |a_k|[/tex]
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