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MathGroup Archive 2006

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Re: How to explain unexpected behaviour of Fibonacci[]?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70477] Re: How to explain unexpected behaviour of Fibonacci[]?
  • From: "ben" <benjamin.friedrich at gmail.com>
  • Date: Tue, 17 Oct 2006 02:59:12 -0400 (EDT)
  • References: <egvbbm$rrm$1@smc.vnet.net>

Dear Peter,

It seems as if the weird behavior of Limit together with Fibonacci has
something
to do with the way Fibonacci is defined in terms of Hold and
ConditionalHold.
FunctionExpand resolves the problem.

The following works fine,

\!\(Limit[
    FunctionExpand[Fibonacci[n]\/Fibonacci[n - 1], n \[Element]
Integers],
    n -> Infinity]\)

but we can fool mathematica by

In[36]:=
Limit[Fibonacci[n]/n, n -> Infinity]
0

Out[36]:=
Limit[Fibonacci[n],n\[Rule]Infinity]

Out[37]=
Hold[Hold[
    Hold[Hold[
        Hold[Hold[$ConditionHold[$ConditionHold[
                Hold[Hold[
                    Hold[Hold[
                        Hold[Hold[$ConditionHold[$ConditionHold[
                                Hold[Hold[
                                    Hold[$ConditionHold[$ConditionHold[
                                        Hold[Fail]]]]]]]]]]]]]]]]]]]]]]
Limit[Fibonacci[n],n\[Rule]Infinity]

What still puzzles me is that

Limit[Log[FunctionExpand[Fibonacci[n], n \[Element] Integers]]/n,
  n -> Infinity]

is ok, but moving FunctionExpand outside of Log spoiles everything.

Bye
Ben

> Dear group,
>
> today I came across a strange behaviour either of Limit[] or of Fibonacci[];
>
> Limit[Log[Fibonacci[n]]/n, n -> Infinity]
> --> 0
> This is wrong.
>
> NLimit[Log[Fibonacci[n]]/n, n -> Infinity]
> --> 0.4812124345417209
> is an approximation
>
> and to let Mathematica do all the work:
>
> Limit[Log[SeriesTerm[FullSimplify[
>   PowerSum[Fibonacci[n], {x, n}]], {x, 0, n}]]/n, n -> Infinity]
> --> Log[(1/2)*(1 + Sqrt[5])]
>
> N[%]
> 0.48121182505960347
>
> I would expect an unevaluated Limit[Log[Fibonacci[n]]/n.n->Infinity] or
> Log[GoldenRatio].
> 
> Peter


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