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Stuck on some proofs

  1. Jul 2, 2005 #1
    can anyone help me with the proofs:


    if [tex]F_i[/tex] is the ith Fibonacci number, then





    I proved the last four using induction. But how can i prove them without using induction?
  2. jcsd
  3. Jul 3, 2005 #2


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    Homework Helper

    euler found your sum zeta(2) by writing
    sin(x)=x(1-x^2/(k pi)^2)(1-x^4/(k pi)^4)(1-x^6/(k pi)^6)...
    and expanding the product and setting it equal x-x^3/3!+x^5/5!+...
    It is often done as a routine exercise with fourier series.
    There are many ways to prove it.
    for the other 4 whats wrong with induction?
    where a=.5(1+sqrt(5)) b=.5(1-sqrt(5))
    so you could express those sums as geometric series among other methods
  4. Jul 4, 2005 #3
    there's nothing wrong with induction. but i wanted get the right hand sides from the left hand sides (from scratch), if you know what i mean.

    anyway, thanks very much for the help. and additional information is always welcome.
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