A series for Fibonacci numbers.

Question

How can I prove The Fibonacci sequence is encoded in the number $1/89$ i.e. $( 1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034 \ldots)$

Answer 1

If there were such $x$ ($ = 0.011235\ldots$), then this satisfies the property of Fibonacci sequence: $$\begin{array}{lllllllll} &0.&1&1&2&3&5&\ldots&//\ 10x\\ +&0.&0&1&1&2&3&\ldots&//\ x\\ \hline &0.&1&2&3&5&9&\ldots\\ +&1\\ \hline &1.&1&2&3&5&9&\ldots&//\ 100x \end{array} $$

So by solving for such $x$, $$\begin{align*} 10x + x &= 100x - 1\\ 89x &= 1\\ x &= \frac{1}{89}\end{align*}$$

Answer 2

I like peterwhy’s answer — think of this as an extended comment on it — but it does slide over a potentially tricky point: $x$ isn’t

$$\begin{array}{lll} &0.&0&1&1&2&3&5&\color{brown}8&\color{brown}13&\ldots\;, \end{array}$$

as the picture might suggest, but rather

$$\begin{array}{lll} &0.&0&1&1&2&3&5&\color{brown}9&5&\ldots\;, \end{array}$$

because of carries. We actually have

$$x=\sum_{n\ge 1}\frac{F_{n-1}}{10^n}\;.$$

Now we can carry out peterwhy’s calculation as follows:

$$\begin{align*} 0.01+0.1x+0.01x&=0.01+\sum_{n\ge 1}\frac{F_{n-1}}{10^{n+1}}+\sum_{n\ge 1}\frac{F_{n-1}}{10^{n+2}}\\\\ &=0.01+\sum_{n\ge 2}\frac{F_{n-2}}{10^n}+\sum_{n\ge 3}\frac{F_{n-3}}{10^n}\\\\ &=0.01+\frac{F_0}{10^2}+\sum_{n\ge 3}\frac{F_{n-2}+F_{n-3}}{10^n}\\\\ &=0.01+\sum_{n\ge 3}\frac{F_{n-1}}{10^n}\\\\ &=\sum_{n\ge 1}\frac{F_{n-1}}{10^n}\\\\ &=x\;, \end{align*}$$

so $0.01=0.89x$, and $x=\dfrac1{89}$.