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A logarithmic convergence tests - Analysis

  1. May 8, 2014 #1
    1. The problem statement, all variables and given/known data

    Given a non-negative sequence [itex]\{a_{n}\}_{n=1}^{\infty}[/itex]. Proove that the serie [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] converge if and only if [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n})[/itex] converges.



    2. Relevant equations



    3. The attempt at a solution

    My first attempt is the direct comparison test. Which tells that if [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n})[/itex] converges and if there exist a reel number γ so [itex]a_{n}\leq\gamma\ln(1+a_{n})[/itex] for all n, then the series [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] also converges.

    But i can't find such a γ.
     
    Last edited: May 8, 2014
  2. jcsd
  3. May 8, 2014 #2
    Hint*: Note that ##\sum_{n=1}^\infty ln(1+a_n)## is a sum of logarithms. What can you do to change logarithms into multiplication?

    *##log(a)+log(b)=log(ab)##.
     
  4. May 8, 2014 #3
    yeah, so [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n}) = \ln\Big[\Pi_{n=1}^{\infty}(1+a_{n})\Big][/itex]. But how am I supposed to conclude from this, that the series [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] must then also converge ?
     
  5. May 8, 2014 #4
    Well we know that ##\sum_{n=1}^\infty ln(1+a_n)=C##, where C is a constant, since the sum converges. And we know ##\sum_{n=1}^\infty ln(1+a_n)=ln[\prod_{n=1}^\infty(1+a_n)]=C##. So if we take the exponent of both sides we get ##\prod_{n=1}^\infty(1+a_n)=e^C##. If you expand the product what do you notice? Now you have a bound for the sum ##\sum_{n=1}^\infty a_n ##
     
  6. May 8, 2014 #5

    pasmith

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    Homework Helper

    Why is that if every [itex]a_n \geq 0[/itex], we have that [tex]\exp\left(\sum_{n=1}^N \ln(1 + a_n)\right) = \prod_{n=1}^N (1 + a_n) \geq 1 + \sum_{n=1}^N a_n \geq 1
    [/tex] for every [itex]N \in \mathbb{N}[/itex]? (Hint: consider what happens if you expand the product.)

    If [itex]\sum_{n=1}^\infty \ln(1 + a_n)[/itex] converges, what can you say about the convergence of [itex]\sum_{n=1}^\infty a_n[/itex]?

    Conversely, if [itex]\sum_{n=1}^\infty a_n[/itex] diverges, what can you conclude about the convergence of [itex]\sum_{n=1}^\infty \ln(1 + a_n)[/itex]?
     
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