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hagman
Posts:
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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


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



