I've just learned about e. I am very much the novice and my problem is that while trying to calculate the convergent fractions for e. For instance:
$${2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4}}}}}$$
I end up with 144/53?
I was wondering are there specific steps that I'm missing? For me I've been starting at the end of the continued fraction and working my way left. For instance:
$\frac{3}{1} + \frac{3}{4}$
And get 15/4 and then:
$\frac{2}{1} / \frac{15}{4}$
Until I finish with 144/53, which I'm not seeing this anywhere as one of the first few convergents of e.
You’re using a generalized continued fraction; the convergents that you normally see listed are those for the standard continued fraction expansion of $e$, i.e., the one with $1$ for each numerator:
$$e=[2;1,2,1,1,4,1,1,6,1,1,8,\dots]\;.$$
This can also be written
$$[1;0,1,1,2,1,1,4,1,1,6,1,1,8,\dots]$$
to emphasize the pattern even more strongly.