$$A = a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cdots}}},$$ where $a_n = f(n), f:\mathbb{Z^+} \rightarrow \mathbb{Z^+}$.
Is there an easy way of calculating the value of $A$ using algebra and/or simple calculus for a given $f(n)$? And are all such fractions convergent?