Prove that the infinite sum $\sum_{n=1}^{\infty} \frac{F_{n}}{ 10 ^ n }$ converges to a rational number

Solution 1:

We have $F_n=\frac{\varphi^n-\psi^n}{\sqrt5}$. And using a geometric sums we get $$\sum_{n=1}^\infty\frac{F_n}{10^n}=\frac1{\sqrt 5}\sum_{n=1}^\infty\frac{\varphi^n}{10^n}-\frac1{\sqrt 5}\sum_{n=1}^\infty\frac{\psi^n}{10^n}=\frac1{\sqrt5}\frac{\frac\varphi{10}}{1-\frac{\varphi}{10}}-\frac1{\sqrt5}\frac{\frac\psi{10}}{1-\frac{\psi}{10}}=\frac{40}{(19+\sqrt5)(19-\sqrt5)}=\frac{10}{89}$$

Solution 2:

What happens if you take the Fibonacci sequence and subtract the sequence shifted by one?

  1 1 2 3 5 8 13 21 34 …
- 0 1 1 2 3 5  8 13 21 …
= 1 0 1 1 2 3  5  8 13 …

You get the sequence shifted by two, except for the first number. Now let's introduce a “decimal point”, even though the numbers are not really digits of the decimal expansion. You may think of this as a formal sum if you prefer doing things the formal way, but a decimal point feels more intuitive to me.

  1.1 2 3 5 8 13 21 34 …
- 0.1 1 2 3 5  8 13 21 …
= 1.0 1 1 2 3  5  8 13 …

The second row is your number $x$. The first row is shifted one digit to the left so it is $10\,x$. The last row is shifted one digit to the right and has one added to it. So it is $1+x/10$. Now you have an equation:

\begin{align*} 10\,x - x &= 1 + x/10 \\ 9\,x &= 1 + x/10 \\ 90\,x &= 10 + x \\ 89\,x &= 10 \\ x &= 10/89 \end{align*}

The choice of where to put the decimal point is pretty arbitrary, you could also have placed it one digit further right and written the equation as $100\,x - 10\,x = 10 + x$, using the fractional part of the third row as your unshifted number.

The above argument works without any need to look up more than the basic definition of the Fibonacci numbers. But it does rely on the convergence of the series. If you don't take that for granted, have a look at the quotient between two consecutive sequence elements.

$$\frac{a_{n+1}}{a_n}=\frac{10^n\,F_{n+1}}{10^{n+1}\,F_n}= \frac1{10}\left(1+\frac{F_{n-1}}{F_n}\right)<\frac{2}{10}<1$$

so by the ratio test the series is absolutely convergent. This justifies that treating the infinite sum like a single number makes sense.

Solution 3:

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 $y>\phi$.