One Over Eighty-nine

As was shown in the last section, multiplying the Fibonacci numbers by increasing powers of 1/10 and adding them together results in a number that looks remarkably like the value 1/89.

In this section, we will attempt to prove that the number you get when you add all these values is, in fact, 1/89. We will do this by setting up a function to add up all the values. First, let's say we have a function f(x), where

Note that this is not a completely arbitrary function. Looking back at this representation of our pattern,

it becomes evident that these numbers are actually the values you get when you plug 1/10 into the function f(x). In other words, we are trying to prove that

The first step in proving this is to reduce f(x) to a simpler, but equal, function. First, we will write the function in open form.

Unfortunately, it's very difficult to deal with a function with an infinite number of terms. But what if we could eliminate all but a few of the terms of the function? The only way to do this is to subtract at least one more infinite functions from it. But what functions should we use?

The coefficients of our function are all members of the Fibonacci series, and we know that every Fibonacci number is the sum of the two Fibonacci numbers that precede it. Therefore, in order to subtract away each coefficient in f(x), we must use functions where the Fibonacci coefficients are offset such that they will subtract away.

The two functions we are looking for are xf(x) and x²f(x).

We mean to subtract these two functions from our original function f(x), so we set up the subtraction. We list like terms underneath each other so we can more easily subtract.

Notice that after the first two columns, the first number is cancelled out by the two below it. This is because each Fibonacci number is the sum of the two Fibonacci numbers that precede it. That is why we picked those two functions the way we did. This will happen in every single successive column, leaving only the first two columns not cancelled.

Also, as an added bonus, the second column cancels out as well. This leaves us with the following equation.

Since what we're looking for is an new value for f(x), we'll factor it out on the left side of the equation.

By solving the equation for f(x), we get

If we simplify this a bit further, we obtain the following expression for f(x).

Recall that the entire point of the proof thus far has been to find a new way to express f(x) so that we could find f(1/10). We now have an expression for f(x), so all we need to do is plug 1/10 into our new equation and solve.

And what do you know? We came out with the number 1/89. So in fact, multiplying the Fibonacci series by increasing powers of 1/10 and adding them does give the value 1/89.