How does one find the value of
$$1+\cfrac 1 {2+\cfrac 2 {3+\cfrac{3}{4+\cfrac{4}{5+\cfrac{5}{\ddots}}}}} \ \text{ or }\ 1+\cfrac{2}{3+\cfrac{4}{5+\cfrac{6}{7+\cfrac{8}{9+\cfrac{10}{\ddots}}}}}$$
Is there any way to find a continued fraction that is not necessarily periodic, but has a definite pattern to it?
Best Answer
There are several answers, but to your obvious question, the first c.f. value is $\;1/(e-2)\approx 1.392211\;$ given by the OEIS sequence A194807 and the second is $\;1/(\sqrt{e}-1)\approx 1.541494\;$ given by the OEIS sequence A113011. For your second question, there are other examples such as in the Wikipedia article Gauss's continued fraction and various q-continued fractions such as Wikipedia article Rogers-Ramanujan and MSE question 1347034