[seqfan] Re: Fibonacci-sums

[Robert Israel]

On Wed, 18 Aug 2010, Gottfried Helms wrote:

> Reading the wikipedia-article on fibonacci-numbers
> I tried what would be the alternating sum of
> all fibonacci-numbers. The sequence is diverging,
> so one has to employ a method of divergent summation.
>
> Happily the growthrate is not too strong, so simple
> Euler-summation suffices.
>
> Using fib(0)=0,fib(1)=1, fib(2)=1, fib(3)=2,... I got
>
>  inf
>  sum    (-1)^k * fib(k)   = -1     // Eulersummation
>  k=0
>
> Well, it is interesting, that it gives such a simple value.

Note that fib(k) = (phi^k - (-1/phi)^k)/sqrt(5), where
phi = (sqrt(5)+1)/2.
So formally
sum_{k=0}^infty (-1)^k fib(k) = (1/(1 + phi) - 1/(1 - 1/phi))/sqrt(5)
which simplifies to -1.

Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada