Alternating series: And alternating series is one in which every other term is negative. This geometric series alternates: S=1/2-1/4+1/8-1/16+... Every alternating series, in which the terms get closer and closer to zero (as a limit), converge. So alternating series are especially well-behaved. Since this one is a geometric series, we can easily calculate the sum using the formula. An alternating harmonic series would converge: S=1-1/2+1/3-1/4+1/5-... This sum is 0.693147180559945.... This is a famous alternating series: pi/4=1-1/3+1/5-1/7+1/9-..., and is one of the inefficient ways to calculate pi.
With an alternating series, if you have two consecutive partial sums (like Sn and Sn+1), then the actual sum (of infinitely many terms) is between these partial sums. So, not only does a partial sum give an estimate of the actual sum, but it also helps you to determine just how accurate it is.
How about a series like this: S=1/2+1/4-1/8-1/16+1/32+1/64-... (the signs go ++--++--...)? It is not an alternating series, and it is not a geometric series. But it is the sum of two alternating (and geometric) series: 1/2-1/8+1/32-... + 1/4-1/16+1/64, so it converges. Any series which goes ++--++--... can be split up into two alternating series.
>
> I believe this gentlemen is trying to suggest that this equation is more or less totally accurate rather than a approximation. I believe the latter, myself.
>
> As I mentioned before, mathematicians have used programs with millions of numbers in order to solve the mystery of Pi but without success.
>
> Therefore, I fail to see how this version he mentions can be spot-on rather than a close approximation.
> ----- Original Message -----
> From: <[EMAIL PROTECTED]>
> To: "U.S. Metric Association" <[EMAIL PROTECTED]>
> Cc: <[EMAIL PROTECTED]>
> Sent: Monday, June 14, 2004 8:47 PM
> Subject: [USMA:30115] Re: Liebnitz and Pi.
>
>
> > Hello Stephen!
> >
> > The "fractionalization" of pi is not at all part of a scheme by anti-metric folks. Fractions were used as rough APPROXIMATIONS of pi, (i.e. 3 1/7) in the past. With respect to the fractional series provided at the bottom of your message, that is a good series for approximating pi (given that a sufficient number of fractions).
> > As someone who knows the history of the development of this magnificent number, I can tell you that there is NO reason for me to believe that BWMA, metric-sucks, freedom2mearure-like people are behind this "fractinaliztion 'attempt'" of pi.
> >
> > Hope this helps!
> >
> > METRIC ROCKS!
> >
> > -----Thanks!-----
> >
> > Cole Kingsbury
> > USMA member - Age 19
> > [EMAIL PROTECTED]
> >
> > -------------------
> >
> >
> >
> > > Can anyone on this mailing list confirm something for me? I had been arguing
> > > the benefits of decimals and metric over fractions and imperial measurements
> > > with someone on another website and Pi was mentioned.
> > >
> > > I realised that mathematicians over the years have tries to solve Pi by feeding
> > > millions of decimal numbers into computers but to no avail.
> > >
> > > Then he mentioned Liebnitz who, apparently, according to him, had solved Pi
> > > using fractions in the 1670's.
> > >
> > > What I would like to know is, is this actual fact or just another misleading
> > > piece of propaganda put about by the anti-metric brigade?
> > >
> > > The equation he mentioned is below:
> > >
> > > he riddle had of course been solved by Leibnitz in the 1670s with a continuous
> > > series of fractions:
> > >
> > >
> > > Pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17 - 1/19 etc...)
> > > or expressed in C to 2^24 this code is amazingly accurate.
> > >
> > >
> > > int x = 0;for (int i = 1; i < 16777216; i++) { x += 1/((4*i)-3); x -=
> > > 1/((4*i)-1); }int pi = 4 * $x;cout pi;
> > >
> >
>
>
