I am trying to evaluate an infinite series that involves Fibonacci numbers. By adding the first thousands of terms, I get an approximation of the sum: $S = 5.3598856662431775531720113029189271796889051337319684864955538153...$.
I guess that $S$ can be written exactly as $a+b\sqrt{5}$, where $a, b \in \mathbb{Q}$. Suppose for the moment that it is true. To find $a$ and $b$, I wonder if there are any methods similar to finding convergents of continued fractions. Of course, if we suppose $a=0$, then we can find $b \in \mathbb{Q}$ that approximates $\frac{S}{\sqrt{5}}$ as well as we wish. But here I am allowing one more free variable in the hope that the approximation is in fact exact, i.e. $S = a+b\sqrt{5}$.
Edit: To @Lubin, and in case anyone is wondering, $$S = \sum_{k=2}^\infty \frac{F_{k+2}}{F_{k+1}F_{k-1}},$$ where $F_k$ is the $k$th Fibonacci number, with $F_0=0, F_1=1, F_2=1, ...$.
p.s. I am asking this question as a general approximation question. As suggested by @Bonnaduck (and I've also given a proof in the comments), my guess above is very likely to be false because $S$ involves the reciprocal Fibonacci constant.
The main problem is better phrased as: Given a real number $x$ in decimal form (we have a way to get as many digits as we need), find $a, b \in \mathbb{Q}$ with small denominators such that $a+b\sqrt{5}$ is a good approximation of $x$, similar to the traditional convergents method. Moreover, if $x$ is indeed of the form $a+b\sqrt{5}$, then this method should produce such $a$ and $b$ within finite steps.