<s>[INST] Prove that the infinite sum $\sum_{n=1}^{\infty} \frac{F_{n}}{ 10 ^ n }$ converges to a rational number How do you prove that the following infinite sum
\begin{align}
     &0.1
\\+\;&0.01
\\+\;&0.002
\\+\;&0.0003
\\+\;&0.00005
\\+\;&0.000008
\\+\;&0.0000013
\\ \;&\quad\vdots
\end{align}
converges to a rational number?
Notice that the above sum can be written as 
$$\sum_{n=1}^{\infty} \frac{F_{n}}{ 10 ^  n }$$
where $F_{n} $ is a Fibonacci sequence.
 [/INST] Hint :
Denote the given problem as $x$ and from the recurrence relation $F_{n}=F_{n-1}+F_{n-2}$, we have
$$
\begin{align}
x&=\sum_{n=1}^{\infty} \frac{F_{n}}{10 ^ n}\\
&=\frac{F_1}{10}+\sum_{n=2}^{\infty} \frac{F_{n}}{10 ^ n}\\
&=\frac{1}{10}+\sum_{n=2}^{\infty} \frac{F_{n-1}+F_{n-2}}{10 ^ n}\\
&=\frac{1}{10}+\frac{1}{10}\sum_{n=2}^{\infty} \frac{F_{n-1}}{10 ^{n-1}}+\frac{1}{10^2}\sum_{n=2}^{\infty} \frac{F_{n-2}}{10 ^{n-2}}\\
&=\frac{1}{10}+\frac{1}{10}\sum_{n-1=1}^{\infty} \frac{F_{n-1}}{10 ^{n-1}}+\frac{1}{10^2}\sum_{n-2=1}^{\infty} \frac{F_{n-2}}{10 ^{n-2}}\\
&=\frac{1}{10}+\frac{1}{10}\sum_{n=1}^{\infty} \frac{F_{n}}{10 ^ n}+\frac{1}{100}\sum_{n=1}^{\infty} \frac{F_{n}}{10 ^ n}\\
x&=\frac{1}{10}+\frac{x}{10}+\frac{x}{100}\\
\end{align}
$$
The rest is a piece of cake. In fact, using the same technique, you can generalize the problem into something like this:
$$
\sum_{n=1}^{\infty} \frac{F_{n}}{y ^ n}=\frac{y}{y^2-y-1}
$$
It holds provided $y>\phi$.
</s>