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Topic: 2 approximations and 1 exact formula for pi
Replies: 10   Last Post: Jan 28, 2011 10:59 PM

 Messages: [ Previous | Next ]
 apovolot@gmail.com Posts: 104 Registered: 9/4/08
Re: 2 approximations and 1 exact formula for pi
Posted: Jan 22, 2011 2:36 PM

On Jan 22, 11:57 am, Jacob <a...@hsu.edu> wrote:
> On Fri, 21 Jan 2011 08:15:47 -0800 (PST), plouffe wrote:
>

> >Hello everybody, (this message needs FIXED font).
>
> > these days I am working on exponential sums and I
> >have found something of interest,
> >like
> >                            infinity
> >                             -----          3
> >                              \            n
> >                               )     ---------------
> >                              /          2 Pi n
> >                             -----   exp(------) - 1
> >                             n = 1         13

>
> >= 119.00000000000000000000000000000009593745851025547335588584913...
>
> >the precision is 31 digits.
>
> > Another one is
> >                            infinity
> >                             -----          3
> >                              \            n
> >                               )     ---------------
> >                              /          2 Pi n
> >                             -----   exp(------) - 1
> >                             n = 1         7

>
> > =
> >10.0000000000000001901617678886626755843593058554453334802548978434061099438\
> >The precision here is 15 digits.

>
> >For an argument of 2*Pi*n/163, the precision is 435 digits! I
> >don't know why I took 163... (of course).

>
> >in general, these sums will be near an integer if the argument
> >is 2*Pi*n/k and k is NOT a multiple of 2,3 or 5, strange isn't ?
> >These are the simplest I could find,
> >if the exponent is of the form 4m-1 then the sum is often an
> >integer with the same conditions. This, I believe extends
> >a little bit the known formula of Ramanujan/Berndt/etc.

>
> > A good question is : does someone has a simple explanation
> >of this ? I don't. I am preparing a paper on these results.

>
> >Because, I do have another one which is EXACT, namely for pi:
> >In general these sums with a fractional exponent are very
> >close to integers or pi, I have a couple of series for 1/pi, 1/pi^2,
> >etc.

>
> >here is the  EXACT formula with fractional arguments mixed with
> >integer exponents.

>
> >   /infinity                  \      /infinity                    \
> >   | -----                    |      | -----                      |
> >   |  \              1        |      |  \               1         |
> >10 |   )     -----------------| - 40 |   )     -------------------|
> >   |  /      n (exp(Pi n) - 1)|      |  /      n (exp(2 Pi n) - 1)|
> >   | -----                    |      | -----                      |
> >   \ n = 1                    /      \ n = 1                      /

>
> >          /infinity                    \      /
> >infinity                  \
> >          | -----                      |      |
> >-----                    |
> >          |  \               1         |      |  \
> >1        |
> >     + 10 |   )     -------------------| - 10 |   )
> >-----------------|
> >          |  /      n (exp(4 Pi n) - 1)|      |  /        /    Pi
> >n     \|
> >          | -----                      |      | -----   n |exp(----) -
> >1||
> >          \ n = 1                      /      \ n = 1     \
> >5       //

>
> >          /infinity                    \      /
> >infinity                    \
> >          | -----                      |      |
> >-----                      |
> >          |  \               1         |      |  \
> >1         |
> >     + 40 |   )     -------------------| - 10 |   )
> >-------------------|
> >          |  /        /    2 Pi n     \|      |  /        /    4 Pi
> >n     \|
> >          | -----   n |exp(------) - 1||      | -----   n |exp(------)
> >- 1||
> >          \ n = 1     \      5        //      \ n = 1     \
> >5        //

>
> >> evalf(%);
> >3.14159265358979323846264338327950288419716939937510582097494459230781640628\
> >    62089986280348253421170679821480865132823066470938442...
> >well, I verified up to 1000 digits and it holds.

>
> >This is trivial, ? I do not see how.
>
> >If someone has a piece of information on why this exist, I would
> >be glad to ear from it, references, known results, etc.
> >Me, I never saw these kind of formulas before,

>
> > have a good day,
> > Simon Plouffe

>
> Interesting that a relatively short expression yields a number close
> to an integer.
>
>
> Doesn't one need to know the exact value for Pi as input to the right
> hand side of the formula?
>
> If so, in what sense is Pi being approximated?

Yes, I think Simon did not mean to say that first two formulas have to
do with Pi being approximated - I think Simon presented those two as a
very impressive examples of obtaining "near integers" (thus hint to
Ramanujan via mentioning 163 ;-))

Date Subject Author
1/22/11 Jacob
1/22/11 apovolot@gmail.com
1/22/11 w wirkstoff
1/22/11 achille
1/26/11 achille
1/27/11 achille
1/27/11 achille
1/28/11 achille
1/28/11 achille
1/28/11 Brian Q. Hutchings