Is the iterated continued fraction from convergents for Pi/2 equal to 3/2?

• To: mathgroup at smc.vnet.net
• Subject: [mg119757] Is the iterated continued fraction from convergents for Pi/2 equal to 3/2?
• From: Marvin Burns <marburns at umail.iu.edu>
• Date: Mon, 20 Jun 2011 19:38:16 -0400 (EDT)

```Some time ago I asked for help in defining and naming some iterated
continued fraction from convergents. Links describing them are
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 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!

Similar experiments have indicated that the cc of 1<x<2 is the same for all
x in that interval. I think this can be proven through methods found near
the end of https://oeis.org/wiki/Table_of_convergents_constants.

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. =EF=82=87

N[FromContinuedFraction[%],40]

1.499999938132076351161240652364859960316

Convergents[%,30];

ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30
terms. =EF=82=87

N[FromContinuedFraction[%],40]

1.499999996906567523984731477342040062854

Convergents[%,30];

ContinuedFraction::incomp: Warning: ContinuedFraction terminated before 30
terms. =EF=82=87

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

```

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