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Integration, proofs, spline interpolation

7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers.

(a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n!

(b) Make a clear case that the integral exists for any n>-1, whether n is an integer or not. Thus we could define the factorial function by this integral.

(c) Use this new definition to compute 0!

(d) Use spline interpolation on the values of 0!, 1! and 2! to estimate a reasonable value for (1/2)! This means you should find a polynomial An2+Bn+C that agrees with the factorial function at n=0,1,2, and evaluate it at n=1/2.

(e) The exact value of (1/2)! is sqrt(pi)/2. Check that your estimate above was reasonably accurate, and use this fact to find (-1/2)! Make a sketch of the graph of the factorial function on the interval [-1/2,2] based on these values, assuming that factorial is a reasonably smooth function.

I need to show thinking and reasons for each step! Help!

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Since parts (a)-(c) are the ones that require help with "thinking and reasons", they are the ones I'll address.

(a) You meant to write n! = int_(0)^(infty) (x^n * e^(-x) )
Set u = x^n, dv =e^(-x). Then integration by parts says:
int_(0)^(infty) u dv = - int_(0)^(infty) v du + ...

Solution Summary

This solution is comprised of a detailed explanation to answer Integration, proofs, and spline interpolation problems.