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Topic: Integral of a factorial function (Read 14213 times) 

howard roark
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Integral of a factorial function
« on: Jan 7^{th}, 2009, 5:49pm » 
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What is the integral of a factorial function, say f(n)=n!


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ThudnBlunder
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Re: Integral of a factorial function
« Reply #1 on: Jan 7^{th}, 2009, 6:35pm » 
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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.


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howard roark
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Re: Integral of a factorial function
« Reply #2 on: Jan 7^{th}, 2009, 7:57pm » 
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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....


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Henk
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Re: Integral of a factorial function
« Reply #3 on: Nov 5^{th}, 2009, 9:25am » 
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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 ) )


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SWF
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Re: Integral of a factorial function
« Reply #4 on: Apr 13^{th}, 2012, 6:58pm » 
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To estimate S= n! + (n1)! + (n2)! + ..., 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! + (m1)! + (m2)! = m^{2}(m2)! (assuming m>1) S= n! + (n1)^{2}(n3)! + (n4)^{2}(n6)! + (n7)^{2}(n9)! + ... S= n!*{ 1 + (n1)/n/(n2) + (n4)/n/(n1)/(n2)/(n3)/(n5) + (n7)/n/(n1)/(n2)/(n3)/(n4)/(n5)/(n6)/(n8 ) + ... } 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.


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