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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.

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    @RobertMastragostino thanks for the edit, I had just got the correct syntax on when it said you had already corrected it. I'm going off wolfram for one, 2, 3 , 8/3, 11/4, 19/7,....2012-07-12

1 Answers 1

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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.

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    To get the fraction form we take the first number 2 over unity, then the next fraction would be the first period times numerator and denominator; adding unity to the numerator?2012-07-12