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Topic: Series 1 + 1/(1 . 3) + 1/(1 . 3. 5) + 1/(1. 3 . 5. 7) .....
Replies: 2   Last Post: Aug 18, 2012 6:05 PM

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hagman

Posts: 1,923
Registered: 1/29/05
Re: Series 1 + 1/(1 . 3) + 1/(1 . 3. 5) + 1/(1. 3 . 5. 7) .....
Posted: Aug 18, 2012 6:05 PM
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Am Samstag, 18. August 2012 21:28:24 UTC+2 schrieb James Waldby:
> On Sat, 18 Aug 2012 08:13:13 -0700, hagman wrote:
>
>
>

> > Am Samstag, 18. August 2012 15:46:56 UTC+2 schrieb steine...@gmail.com:
>
> >> That's what I did. I got f'(x) = 1 + x f(x). Then I got stuck. This equation is equivalent to f'(x) - xf(x) = 1. The integrator factor is e^(-x^2/2). Hence, d/dx(e^(-x^2/2) f(x)) = e^(-x^2/2). I can't integrate this.
>
> ...
>

> > With
>
> > erf(x) = 2/sqrt(pi) * int_{-oo}^x exp(-t^2/2) dt

This should read

erf(x) = 2/sqrt(pi) * int_0^x exp(-t^2/2) dt

to ensure erf(0)=0.

>
> > we have
>
> > erf'(x) = 2/sqrt(pi) * exp(-x^2/2)
>
> > Hence with
>
> > f(x) := sqrt(pi)/2 * exp(x^2/2)*erf(x),

This is where we need erf(0)=0 so that f(0)=0.

>
> > we have
>
> > f'(x) = sqrt(pi)/2 * x*exp(x^2/2) * erf(x) + 1,
>
> > i.e.
>
> > f'(x) = x*f(x) + 1.
>
> >
>
> > We are looking for f(1), i.e.
>
> > sqrt(pi)/2 * exp(1/2)*erf(1) = sqrt(e pi)/2 * erf(1)
>
> > hence your original guess is off by a factor of
>
> > erf(1) ~~ 0.8427
>
> ...
>
>
>
> I think there's some problem there, as the sum
>
> 1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) +... is about 1.410686
>
> while sqrt(e*pi)/2 is about 1.461141, which gives a multiplicative
>
> factor of either 1.036 or inversely 0.965, neither being close
>
> to 0.8427.
>
>
>
> --
>
> jiw


You are right - but so was I, except that I managed to lookup in the wrong errfunc table and use the wrong definition of error function in the first place. :)
(Note that 2 * 0.9654... - 1 = 0.8427...)

I have no idea why I didn't simply use PARI to calculate
> intnum(X=0,1,exp(-X^2/2))*2/sqrt(Pi)
0.96546873866986730832269531128090862149

With the correct definition of error function, i.e.
errfunc(x) := 2/sqrt(pi) * int_0^x exp(-t^2) dt
the correct result should be something like
1/1 + 1/(1*3) + 1/(1*3*5) + 1/(1*3*5*7) + ...
= sqrt(e*pi)/2 * errfunc(1/sqrt(2))*sqrt(2)
= sqrt(2*e) * int_0^sqrt(1/2) e^(-x^2) dx
~~ 1.410686134642447997690824711419115
where errfunc(1/sqrt(2)) ~~ 0.682689492137085897170465091264
is the famous number for the "plus/minus one sigma" interval
of normal distributions.

Sorry for the mess I caused.

hagman



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