--- a hazardous summation-method for 0!-1!+2!-3!+...-... - sci.math

Message from discussion --- a hazardous summation-method for 0!-1!+2!-3!+...-...

G. A. Edgar

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More options Nov 7 2009, 10:52 am

Newsgroups: sci.math

From: "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>

Date: Sat, 07 Nov 2009 05:52:41 -0500

Local: Sat, Nov 7 2009 10:52 am

Subject: Re: --- a hazardous summation-method for 0!-1!+2!-3!+...-...

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In article <7lkkp5F3e45d...@mid.dfncis.de>, Gottfried Helms

<he...@uni-kassel.de> wrote:
> Am 06.11.2009 23:43 schrieb Gottfried Helms:
> > What do you think about the approach to the divergent summation of
> > the series in the header

> > su = 0! - 1! + 2! - 3! + 4! - ... + ...

> > L Euler found a meaningful interpretation using integrals assigning
> > it a value of about 0.596347…
> > Also the Borel-summation assigns the same value to this.

> I propose the following further values for variations of that sum:
> fsum(x) = 0! + 1! x + 2! x^2 + 3! x^3 +…

> fsum(-1) = 0.596347362323
> fsum(-2) = 0.461455316242
> fsum(-3) = 0.385602012137
> fsum(-4) = 0.335221361210
> fsum(-5) = 0.298669749329
> fsum(-6) = 0.270633013639
> fsum(-7) = 0.248281352547
> fsum(-8) = 0.229947781627
> fsum(-9) = 0.214577094581
> fsum(-10) = 0.201464233646
> fsum(-11) = 0.190117766778
> fsum(-12) = 0.180183310425
> … …

> Could someone check this using the integral-formula?

> Gottfried Helms

According to Borel summation, we should have

-exp(1)*Ei(-1) = .5963473622
-(1/2)*exp(1/2)*Ei(-1/2) = .4614553164
-(1/3)*exp(1/3)*Ei(-1/3) = .3856020120
-(1/4)*exp(1/4)*Ei(-1/4) = .3352213612
-(1/5)*exp(1/5)*Ei(-1/5) = .2986697494
-(1/6)*exp(1/6)*Ei(-1/6) = .2706330136
-(1/7)*exp(1/7)*Ei(-1/7) = .2482813514
-(1/8)*exp(1/8)*Ei(-1/8) = .2299477818
-(1/9)*exp(1/9)*Ei(-1/9) = .2145771028
-(1/10)*exp(1/10)*Ei(-1/10) = .2014642544
-(1/11)*exp(1/11)*Ei(-1/11) = .1901177930
-(1/12)*exp(1/12)*Ei(-1/12) = .1801833179

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

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