From an exercise in Burton's number theory text:
Given the infinite continued fraction [1;3,1,5,1,7,1,9,...], find the best rational approximation $a/b$ with denominator $b < 25$
Calculating the convergents, I have 1, 4/3 , 5/4 , 29/23, 34/27 , ...
I know that since 29/23 is a convergent, it is within $1/23^2$ of the value of the infinite fraction and is the best approximation with a denominator less than or equal to 23.
However, I don't know that there isn't a rational approximation better than 29/23 with denominator of 24 in a straightforward manner.
My approach is to observe that $30/24 < 29/23 - 1/23^2$ and $31/24 > 29/23 + 1/23^2$ so no rational with denominator 24 is a better approximation than 29/23.
It seems like there should be a cleaner more general approach?