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Convergence/Divergence Proof

  1. Mar 3, 2014 #1
    1. The problem statement, all variables and given/known data

    Show if an > 0 for each n[itex]\in[/itex][itex]N[/itex] and if ∑an converges, then ∑an2 converges and that ∑1/an diverges.

    NB. all ∑ are between n=1 and ∞

    2. Relevant equations



    3. The attempt at a solution

    Let partial sums of ∑an2 be Sk = a1 + ... + ak

    To say ∑an2 is absolutely convergent is to say ∑|an2| is convergent.

    It follows that partial sums Tk = |a12| + |a22| + ... + |ak2| of the series are bounded above by M.

    Then by an extended form of the Triangle Inequality we have:

    |Sk|

    = a12 + a22 + ... + ak2

    ≤ |a12| + |a22| + ... + |ak2|

    = Tk

    ≤ M

    Hence the sequence {|Sk2} is bounded above by M. It is bounded below by 0 as an > 0 so an2 > 0. Therefore it is bounded.

    It is therefore convergent.

    Is this correct so far? Would a similar proof follow for showing ∑1/an is divergent?

    Can anyone help with this please? :redface:
     
  2. jcsd
  3. Mar 3, 2014 #2

    PeroK

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    I think you're missing a simple solution. Think of what happens to a_n as n gets large.

    I don't follow your proof. (How are you using the convergence of the series a_n?)
     
  4. Mar 3, 2014 #3

    micromass

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    What is ##M## and why does it bound the ##T_k##?
    You have only established that ##\sum a_n^2## being absolutely convergent is the same as saying that ##\sum|a_n|^2## converges. But you have not established the fact that ##\sum a_n^2## actually absolutely converges!
     
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