MathGroup Archive 2001

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Formula for sum of integrals of polynomials

Its been a week since I started using mathematica. My purpose has been
to explore the following expression. My ultimate aim is to look at its
limit as n tends to infinity, although to know its closed form
expression would be nice. The following code gives me the numerical
values for particular values of n.

How can I get a closed form expression in terms of n?
Or atleast a closed form expression in terms of n and k for the
general integrand? As you might notice the integrands are "nice"
polynomials ( in x with parameters n and k) raised to the power 2^n.
Can mathematica help me "simplify" these? ( Look at f[k_,n_,x_] and
g[k_, n_, x_] in the p.s.)

closed[n_] := 
        1 - (Sum[((((-1)^i)*Binomial[n, i]*(x - i)^n)/n!)
                 ,{i,0,k}])^(2^n), {x, k, k + 1},
        GenerateConditions -> False], {k, 0, n - 1}];

Any help or suggestions about how to tackle this problem would be
greatly appreciated.

Hemanshu Kaul

p.s. I have already tried breaking this expression into smaller parts
( written recursively ) but that didn't work properly. The code I
wrote was -

Clear[f, g, h, closed, k, n, x];

f[0, n_, x_] = x^n/(n!);

f[k_, n_, x_] := 
    f[k - 1, n, x] + (((-1)^(k - 1))*Binomial[n, k - 1]
                                               *(x - k + 1)^n)/n!;

g[k_, n_, x_] := f[k, n, x] ^ (2^n);

closed[n_] = 
        Integrate[1 - g[k, n, x], {x, k, k + 1}, 
                                   GenerateConditions -> False],
                                                   {k, 0, n - 1}];

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