Concrete mathematics : a foundation for computer science

46 SUMS
One way to use this fact is to examine the error in the approximation,
E, = 0, - in3. Since � ,, satisfies the recurrence 0, = [7,-l + n2, we find
that E, satisfies the simpler recurrence
En = II,-fn3 = IJP1 +n2-in3 = E,p1+~(n-1)3+n2-3n3
= E,-1 +n-5.
Another way to pursue the integral approach is to find a formula for E, by
summing the areas of the wedge-shaped error terms. We have
n
on -
s 0
x2dx = 2 (k2-/;P,x2dx)
k2 _ k3 - (k- 1)3
Either way, we could find E, and then !I,.
3
= f(k-f)
Method 5: Expand and contract.
Yet another way to discover a closed form for Cl, is to replace the original
sum by a seemingly more complicated double sum that can actually be
simplified if we massage it properly:
= t (F)(n-j+l)
l