Concrete mathematics : a foundation for computer science

The horizontal scale
here is ten times the
vertical scale.
solution when a = 0, p = 1, y = -3, 6 = 3. Hence
3D(n) - 3C(n) + B(n) = n3 ;
2.5 GENERAL METHODS 45
this determines D(n).
We’re interested in the sum Cl,, which equals � -1 + n2; thus we get
17, = R, if we set a = /3 = y = 0 and 6 = 1 in (2.41). Consequently
El, = D(n). We needn’t do the algebra to compute D(n) from B(n) and
C(n), since we already know what the answer will be; but doubters among us
should be reassured to find that
3D(n) = n3+3C(n)-B(n) = n3+3T-n = n(n+t)(n+I),
Method 4: Replace sums by integrals.
People who have been raised on calculus instead of discrete mathematics
tend to be more familiar with j than with 1, so they find it natural to try
changing x to s. One of our goals in this book is to become so comfortable
with 1 that we’ll think s is more difficult than x (at least for exact results).
But still, it’s a good idea to explore the relation between x and J, since
summation and integration are based on very similar ideas.
In calculus, an integral can be regarded as the area under a curve, and we
can approximate this area by adding up the areas of long, skinny rectangles
that touch the curve. We can also go the other way if a collection of long,
skinny rectangles is given: Since Cl, is the sum of the areas of rectangles
whose sizes are 1 x 1, 1 x 4, . . . , 1 x n2, it is approximately equal to the area
under the curve f(x) = x2 between 0 and n.
f(x 1 t
123
I
i
c
n X
The area under this curve is J,” x2 dx = n3/3; therefore we know that El, is
approximately fn3.