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What is the closed form for this?

  1. Oct 27, 2007 #1
    Infinite Sum [n=1] 1/Fibonacci[n]

    Or this,

    Infinite Sum[n=1] 1/(n^n)

    Using some known mathematical constants.
     
  2. jcsd
  3. Oct 27, 2007 #2

    Gib Z

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    Welcome to Physics forums sam! Perhaps you don't exactly know, or you just want to give us problems we can't do to torture us, but:

    The first sum is equal to a known mathematical constant, and its called the Reciprocal Fibonacci Constant:
    [tex]\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots[/tex], which has been proven to be irrational, but it is currently not known if that constant is expressible in terms of more elementary or common constants.

    The second One I know converges, though I have never seen a closed form solution for.
     
  4. Oct 27, 2007 #3
    Hi Gib_Z,

    Sorry, I didn't mean to torture you

    And thanks for the explanation. I was wondering about this the other day and thought what better place to ask about it than here.
     
  5. Oct 27, 2007 #4

    Gib Z

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    This place is about the best place to ask in my opinion as well =] You are really quite lucky, if I hadn't read through the wikipedia article on the Fibonacci numbers just last week, I would not have even remembered about that constant =]
     
  6. Oct 27, 2007 #5

    Gib Z

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    Doing some crawling on the internet I have found out for sure that there is no closed form for the second series, though there is a nice identity that I am attempting at this moment to prove, the derivation isn't given...I should have it by tomorrow, but heres the identity now:

    [tex]\sum_{n=1}^{\infty} \frac{1}{n^n} = \int^1_0 x^{-x} dx[/tex]
     
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