To calculate the convergence of continued fraction, the simplest hand calculation is using bottom up approach where we compute only two elements at the time from the right of condense notation, using formula:
Obtain the approximant of continued fraction
Try to use Continued Fraction Calculator below. Synthesis button converts the continued fraction to into the approximant. Analysis button converts the decimal value into continued frcation.
Continued Fraction Calculator
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html