Iterated continued fraction from convergents are described at https://oeis.org/wiki/Convergents_constant and https://oeis.org/wiki/Table_of_convergents_constants.
Do you think there is any error in the computations, or perhaps in my interpretation, in my new notebook which is pasted to the bottom of this question? It indicates the iterated continued fraction from convergents, or convergents constant (cc), for Pi/2 is 3/2. Can anyone help me prove that Mathematica is correct? The warning statement,"ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30 terms" makes me worry that there could be some error in the computation.
N[Pi/2,40]
1.570796326794896619231321691639751442099
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.399437073110430452143756644740611223113
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.490343396341538312190367098091859766573
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499506141544753996245023837037133508709
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499975256017086378103094352668241530238
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499998762644884859572966908242122674513
Convergents[%,30];
ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30 terms.
N[FromContinuedFraction[%],40]
1.499999938132076351161240652364859960316
Convergents[%,30];
ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30 terms.
N[FromContinuedFraction[%],40]
1.499999996906567523984731477342040062854
Convergents[%,30];
ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30 terms.
N[FromContinuedFraction[%],40]
1.499999999845328929138608048920727157541
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999992275328240194538348680646578
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999999614321523514259986324767680
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999999978495630126428554334079425
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999999430490592898180226181726996
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999990876577511876363531307127180
Convergents[%,30];
N[FromContinuedFraction[%],40]
1.499999999999543690097706342514153778772
l=Pi/2; Table[c=Convergents[l,500];l=FromContinuedFraction[c],{a,200}];N[l,300]
1.499999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999997535967131264972798688959975537466275259