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Infinite sum proof by induction

  1. Nov 12, 2007 #1
    So I am trying to solve a problem.

    Evaluate [tex]\sum_{k=1}^{\infty}\frac{6^{k}}{(3^{k}-2^{k})(3^{k+1}-2^{k+1})}[/tex].

    Essentially, I've boiled it down to this, but I can't quite prove it:
    and the limit of this as n approaches infinity is 2.

    I need to be able to prove that [tex]3^{n+1}-2^{n+1}+6^{n-1}=(3^{n}-2^{n})^{2}[/tex] in order for my induction hypothesis to work, and I'm having trouble for some reason. Help?
  2. jcsd
  3. Nov 12, 2007 #2
    Perhaps beacuse it isn't true, consider n=3 then

  4. Nov 12, 2007 #3
    Wow, I'm sorry. Perhaps that's why I was unable to do it. My notes are all over the place and I guess I kind of lost track of a lot of numbers somewhere.

    I meant to have [tex](3^{n-1}-2^{n-1})(3^{n+1}-2^{n+1})+6^{n-1}=(3^{n}-2^{n})^{2}[/tex]

    Now I'm able to prove it! Thanks anyway.
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