wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> Integral of a factorial function (Message started by: howard roark on Jan 7th, 2009, 5:49pm)

Title: Integral of a factorial function
Post by howard roark on Jan 7th, 2009, 5:49pm
What is the integral of a factorial function, say f(n)=n!

Title: Re: Integral of a factorial function
Post by ThudanBlunder on Jan 7th, 2009, 6:35pm
y = n! is just a series of points in the plane. How can it have an integral?
However, its generalization y = Gamma(x) is a continuous curve except when x = 0,-1,-2,-3, etc.
But even this does not seem to be integrable, at least not by the Online Integrator (http://integrals.wolfram.com/index.jsp).

Title: Re: Integral of a factorial function
Post by howard roark on Jan 7th, 2009, 7:57pm
Actually I want to find asymptotic tight bound for the function

Sum(i=1 to n) (i!)

That was the reason I asked integral of a factorial....

Title: Re: Integral of a factorial function
Post by Henk on Nov 5th, 2009, 9:25am
Use Euler's integral representation of GAMMA(z), Re(z) > 0, and integrate z under the integral sign.

If asked from Maple12: identify( int( GAMMA(x), x = 1 ... 3 ) ) =>  4/3  + 2/7*(e + ln( 2 ) )

Title: Re: Integral of a factorial function
Post by SWF on Apr 13th, 2012, 6:58pm
To estimate S= n! + (n-1)! + (n-2)! + ..., integrating gamma(x+1) is not going to work well because the function increases so rapidly with n.  Just estimating S by n! may be more accurate than integrating gamma.

Since S increases rapidly with n, a good estimate would be to use the largest few terms of S. You can group them by threes by using:
m! + (m-1)! + (m-2)! = m2(m-2)!  (assuming m>1)

S= n! + (n-1)2(n-3)! + (n-4)2(n-6)! + (n-7)2(n-9)! + ...

S= n!*{ 1 + (n-1)/n/(n-2) + (n-4)/n/(n-1)/(n-2)/(n-3)/(n-5) + (n-7)/n/(n-1)/(n-2)/(n-3)/(n-4)/(n-5)/(n-6)/(n-8 ) + ... }

Each additional term is much less than the last. For small n, make sure not use too many terms or you will get a zero in the denominator.