Re: 1/89 and the Fibonacci sequence.



On Wed, 7 Sep 2005, Dan wrote:

> Sorry about the multiple posts but this was
> a bitch to edit!
>
> The mystery of 1/89 and the Fibonacci sequence
>
>
> 1/89 =
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> 01123595505617977528089887640449438202247191/
> ..
> With a continuing decimal expansion that has
> a period of 44.
>
> Then adding the Fibonacci sequence in this manner --
> ***OOPs- fixed below***
> 0112358
> >>>>+13
> >>>>+>21
> >>>>+>>34
> >>>>+>>>55
> >>>>+>>>>89
> >>>>+>>>>144
> >>>>+>>>>>233 Which creates a right
> >>>>+>>>>>>377 one step offset.
> >>>>+>>>>>>>610
> >>>>+>>>>>>>>987
> >>>>+>>>>>>>>1597
> >>>>+>>>>>>>>>2584
> >>>>+>>>>>>>>>>4181
> >>>>+>>>>>>>>>>>6765
> >>>>+>>>>>>>>>>>10946
> >>>>+>>>>>>>>>>>>..... etc.
> ----------------------------------
> 01123595505617977528089887640449438202247191...
> = 1/89?
>
> Will this continue repeating the period of 1/89
> no matter how many fibonacci numbers are added
> in this manner?
>
> If it does, can it be proved?
>
[rest done in a similar way]

A good illustration of "it may help to solve a more general (more
ambitious) problem.

Let F(n) be the n-th Fibonacci number, then the power series

F(1)*x + F(2)*x^2 + F(3)*x^3 + ...

is a function g of x, and it can be summed by elementary means if you
multiply and divide it by

(1 - x - x^2) .

Collect like terms, and obtain

g(x) = x / (1 - x - x^2)

(find the radius of convergence yourself).

Now what does it do when x = 1/10? (You may need to adjust the position of
the decimal point.)

(One solved also a class of other problems: What is the sum, if the base
is not 10 but 3? Or 100? Or any other base from 2 on?)

(One more remark: The letter g stands for "generating function", and you
can google more about it.)

Cheers, ZVK(Slavek).
.