Formula for sum of integrals of polynomials

*To*: mathgroup at smc.vnet.net*Subject*: [mg31173] Formula for sum of integrals of polynomials*From*: hemanshukaul at hotmail.com (H Kaul)*Date*: Tue, 16 Oct 2001 01:18:49 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, 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_] := Sum[ Integrate[ 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_] = Sum[ Integrate[1 - g[k, n, x], {x, k, k + 1}, GenerateConditions -> False], {k, 0, n - 1}];