I'm new to continuous fractions and since I haven't dabbled in mathematics for several years I'm finding it quite difficult to get back on the horse.
I'm trying to find e given: $$e = 2 + \frac{1}{1 + \frac{1}{2+\frac{2}{3}}} $$
I understand that the approximation for e in this situation would be 2.72, but I'm not quite able to come to this conclusion myself.
I looked through previous questions that were similar to my question and was greeted with some pretty in-depth formulas that I couldn't grasp.
Would anyone be able to explain to me how they'd work out the answer? Thanks in advance.
I think you mean continued fractions not continuous fractions.
In a continued fraction, each numerator is divided by the whole subsequent expression. So I think the fraction you mean is $$2+\dfrac{1}{1+\dfrac{1}{2+\dfrac{2}{3}}}=2+\dfrac{1}{1+\dfrac{1}{8/3}} =2+\dfrac{1}{1+\dfrac{3}{8}} =2+\dfrac{8}{11}\ .$$