1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Infinite product converges if and only if sum converges

  1. Dec 2, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex]a_n[/itex] is a sequence of positive numbers. Prove that [itex]\prod_{n=1}^{\infty} (1+a_n)[/itex] converges if and only if [itex]\sum_{n=1}^{\infty} a_n[/itex] converges.


    2. Relevant equations



    3. The attempt at a solution
    I first tried writing out a partial product: [itex]\prod_{n=1}^{N} (1+a_n) = (1+a_1)(1+a_2)\dots(1+a_N) = 1 + \prod_{n=1}^{N} a_n + \sum_{n=1}^{N} a_n + C[/itex], where [itex]C[/itex] is the sum of all the combinations of the [itex]a_i[/itex], such as [itex]a_1a_2a_N[/itex] etc, but that did not really lead me to much. I was given the hint of using the logarithm, but I am not really sure when I would use that. Perhaps when the sum converges, so does [itex]\log(1+a_n)[/itex], though I'm not sure how that relates.
     
  2. jcsd
  3. Dec 2, 2013 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    How does ##\log(1+a_n)## compare with ##a_n##?
     
  4. Dec 2, 2013 #3
    Well, ##\log(1+a_n) < a_n## so it will converge when ##a_n## does?
    Also, if it does converge then wouldn't
    ##\sum_{n=1}^{N} \log(1+a_n) = \log(1+a_1)+\log(1+a_2)+\ldots+\log(1+a_n) = \log(\prod_{n=1}^{N}(1+a_n))## imply that the product would converge?
     
  5. Dec 2, 2013 #4

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, that's right. The function ##\log## is continuous, which justifies this step:
    $$ \log \lim_{N \rightarrow \infty} \prod_{n=1}^{N} (1+a_n) = \lim_{N \rightarrow \infty} \log \prod_{n=1}^{N} (1+a_n)$$
    Then apply the product-to-sum property of the log to get
    $$\log \prod_{n=1}^{N} (1+a_n) = \sum_{n=1}^{N} \log (1+a_n)$$
    The right hand side is smaller than ##\sum_{n=1}^{N} a_n##. Putting it all together and taking limits gives you a proof that ##\log \prod_{n=1}^{\infty} (1+a_n) \leq \sum_{n=1}^{\infty}a_n##. What can you conclude?

    Note that the problem statement says "if and only if", so you still need to prove the implication in the other direction.
     
  6. Dec 2, 2013 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    For 0 < x < 1 we have
    [tex] x -\frac{1}{2} x^2 < \ln(1+x) < x [/tex]
    This gives you valuable information when N is so large that ##a_n < 1 \: \forall n \geq N##.
     
  7. Dec 2, 2013 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    For 0 < x < 1 we have
    [tex] x -\frac{1}{2} x^2 < \ln(1+x) < x [/tex]
    This gives you valuable information when n is so large that ##a_n < 1##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Infinite product converges if and only if sum converges
Loading...