MHB Explore the Fibonacci Sum Mystery

[soroban]

Consider the Fibonacci Sequence: .$0,1,1,2,3,5,8,13,21,34,\,.\,.\,.$

Now consider: .$\displaystyle S \;=\;\sum^{\infty}_{n=0}\frac{F_n}{10^{n+1}} $We have:

. ..$\begin{array}{ccccccccccccccccc} 0.&0&1 \\ &&&1 \\ &&&&2 \\ &&&&&3 \\ &&&&&& 5 \\ &&&&&&& 8 \\ &&&&&&& 1&3 \\ &&&&&&&& 2&1 \\ &&&&&&&&& 3&4 \\ &&&&&&&&&& 5&5 \\ &&&&&&&&&&& 8&9 \\ &&&&&&&&&&& 1&4&4 \\ &&&&&&&&&&&&2&3&3 \\ \hline 0. & 0 & 1 & 1 & 2 & 3 & 5 & 9 & 5 & 5 & 0 & 4 & 6 & 1 & . & . & .\end{array}$The sum happens to be $\dfrac{1}{89}$ . . . How strange is that?

 

[chisigma]

Consider the Fibonacci Sequence: .$0,1,1,2,3,5,8,13,21,34,\,.\,.\,.$

Now consider: .$\displaystyle S \;=\;\sum^{\infty}_{n=0}\frac{F_n}{10^{n+1}} $We have:

. ..$\begin{array}{ccccccccccccccccc} 0.&0&1 \\ &&&1 \\ &&&&2 \\ &&&&&3 \\ &&&&&& 5 \\ &&&&&&& 8 \\ &&&&&&& 1&3 \\ &&&&&&&& 2&1 \\ &&&&&&&&& 3&4 \\ &&&&&&&&&& 5&5 \\ &&&&&&&&&&& 8&9 \\ &&&&&&&&&&& 1&4&4 \\ &&&&&&&&&&&&2&3&3 \\ \hline 0. & 0 & 1 & 1 & 2 & 3 & 5 & 9 & 5 & 5 & 0 & 4 & 6 & 1 & . & . & .\end{array}$The sum happens to be $\dfrac{1}{89}$ . . . How strange is that?

In...

Generating Function -- from Wolfram MathWorld

... the generating function of the Fibonacci's sequence $f_{n}$ is said to be...

$\displaystyle g(x)= \sum_{n=0}^{\infty} f_{n}\ x^{n} = \frac{x}{1-x-x^{2}}$ (1)

Setting $x=\frac{1}{10}$ in (1) You obtain...

$\displaystyle \sum_{n=0}^{\infty} f_{n}\ x^{n} = \frac{10}{89}$ (2)

Kind regards

$\chi$ $\sigma$

 

[chisigma]

In...

Generating Function -- from Wolfram MathWorld

... the generating function of the Fibonacci's sequence $f_{n}$ is said to be...

$\displaystyle g(x)= \sum_{n=0}^{\infty} f_{n}\ x^{n} = \frac{x}{1-x-x^{2}}$ (1)

Setting $x=\frac{1}{10}$ in (1) You obtain...

$\displaystyle \sum_{n=0}^{\infty} f_{n}\ x^{n} = \frac{10}{89}$ (2)

The series...

$\displaystyle \sum_{n=0}^{\infty} f_{n}\ x^{n}$ (1)

... converges for $\displaystyle |x|< \frac{-1 + \sqrt{5}}{2} = .6180339887...$, so that, in my opinion, much more 'suggestive' is the result... $\displaystyle \sum_{n=0}^{\infty} \frac{f_{n}}{2^{n}}= 2$ (2)

http://www.sv-luka.org/ikone/ikone180a.jpg

Marry Christmas from Serbia

Kind regards

$\chi$ $\sigma$