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Re: Approximating the sum of a series

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On 2013-07-23, Axel Vogt <&noreply@axelvogt.de> wrote:
> On 23.07.2013 21:29, mmatica@personal.ro wrote:
>> The problem is not related to the exact sum of the series
>> but about the possibility to approximate such series (directly) in Maple.
>> (there should be some flags/options).

>> BTW, Borwein & Borwein have proved that

>> 1/10^5*sum(exp(-n^2/(10^10)),n=-infinity..infinity)

>> = sqrt(Pi) + e

>> 0< |e| < 10^(-42000000000)


> Ok, I am not Borwein^2 and assuming their result the suggested
> formula can not be correct (BTW: the assumptions for AbelPlana
> are not satisfied).

> For the numerical question I would guess: something similar to
> slowly convergent integrals happens and Maple's accellerators
> can not handle it (not even in high precision), exp(-10^5) is
> too 'close' to one (for larger n).

The Borwein result follow from the Jacobi inversion formula for
a sum of a periodic series.  If 

	f(x) = \sum g(x+n)

where the sum is from -infinity to infinity, then the
periodic function is given by

	f(x) = \sum c_k * exp(2*\pi*i*k*x),

where c_k = \int exp(-2*\pi*i*k*t)*g(t) dt,

the integral going from -infinity to infinity.

This cannot be used in all situations, but for 

	\sum exp(-h*(n+x)^2)

one of the series convergers rapidly.

However, this would not be helpful in many other cases.

Both for sums and integrals, Maple makes a few tries and then gives up if
things do not look like the function is well-behaved, whereas a better
lookahead would find that it is well-behaved after an allowable number
of terms or other device.


-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558

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Approximating the sum of a series mmatica@personal.ro - 2013-07-23 00:48 -0700
  Re: Approximating the sum of a series "Nasser M. Abbasi" <nma@12000.org> - 2013-07-23 07:17 -0500
  Re: Approximating the sum of a series acer <maple@rogers.com> - 2013-07-23 07:01 -0700
    Re: Approximating the sum of a series Axel Vogt <&noreply@axelvogt.de> - 2013-07-23 18:57 +0200
  Re: Approximating the sum of a series mmatica@personal.ro - 2013-07-23 12:13 -0700
  Re: Approximating the sum of a series mmatica@personal.ro - 2013-07-23 12:29 -0700
    Re: Approximating the sum of a series Axel Vogt <&noreply@axelvogt.de> - 2013-07-23 22:11 +0200
      Re: Approximating the sum of a series mmatica@personal.ro - 2013-07-23 14:08 -0700
        Re: Approximating the sum of a series Axel Vogt <&noreply@axelvogt.de> - 2013-07-28 19:40 +0200
        Re: Approximating the sum of a series / an Integral Axel Vogt <&noreply@axelvogt.de> - 2013-07-29 22:22 +0200
          Re: Approximating the sum of a series / an Integral Herman Rubin <hrubin@skew.stat.purdue.edu> - 2013-07-30 18:03 +0000
      Re: Approximating the sum of a series Herman Rubin <hrubin@skew.stat.purdue.edu> - 2013-07-25 23:37 +0000
    Re: Approximating the sum of a series acer <maple@rogers.com> - 2013-07-23 14:02 -0700

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