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MathGroup Archive 1998

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Re: how to compute pi by using continued fraction?



Levasseur <levasseu@bit-net.com> writes:

> Jing S Chen wrote:
> > 
> > Hello:
> >     I'm a student of City College of San Francisco. Right now, I have a
> > project that I've been working on at least 4 days. Yet I still have no
> > idea at all.
> >     The purpose of the project is to compute pi by using continued
> > fraction. Here is the formula.
> > 
> > pi = 3 + ( 1 / ( 7 + ( 1 / ( 7 + ( 1 / ( .... 1 / 7)
> > 
> > that means
> >                      1
> > pi = 3 + -----------
> >                7 + /   1               \
> >                     (   ------            )
> >                      \   7 +            /
> >                            .......
> >                                    1
> >                              +   ----
> >                                     7
> > 
> > Is here any one who has a program that can follow the above formula and
> > can generate pi?
> > 
> > Jing S. Chen
> > e-mail: jchen06@hills.ccsf.cc.ca.us
> 
> Jing:
> 
> You are computing the continued fraction <3,7,7,7,...> which is NOT
> equal to Pi.  If fact any cf that is periodic like has the form a + b
> Sqrt[c].  The continued fraction representation of pi is <3,7,15,1...>
> and has no pattern.  The NumberTheory`ContinuedFraction.m package will

Here is a continued fraction definition of Pi attributed to Ramanujan
that definitely has a pattern:

Pi              1
-- =     --------------
4          1 +    1
               ---------
                    3^2          <---The pattern is odd numbers
               2+ --------
                      5^2
                  2+ -------
                        ...


It takes quite a long time to converge. -- 
----------------------------------------------------------------------
Craig P Earls, LT U.S. Navy 			 cearls@ix.netcom.com MIT Naval
Construction and Engineering           cpearls@mit.edu
----------------------------------------------------------------------



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