Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:


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.

share|cite|improve this question
where is this list of convergent fractions? Typically we force the 'numerators' to be $1$ in standard form, which most lists I've seen use as well. I assume this is why you're getting different answers. – Robert Mastragostino Jul 12 '12 at 2:40
@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,.... – tijko Jul 12 '12 at 2:41
up vote 2 down vote accepted

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:


This can also be written


to emphasize the pattern even more strongly.

share|cite|improve this answer
This is what I had original started with but, somehow veered off course. With this, I would take the integer of the divided fraction for the period and the decimals as the next divisor of 1? I will need to use the generalized continued fraction if I want the numerators to be integers other than 1? – tijko Jul 12 '12 at 2:54
@tijko: Yes, as in this calculation of the first few figures in the continued fraction representation of $\pi$. – Brian M. Scott Jul 12 '12 at 2:58
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? – tijko Jul 12 '12 at 3:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.