Chapter 14 Sums and Asymptotics414
We then apply Theorem 14.3.2 to conclude that
2
3
.n3=2 1/C 1 S
2
3
.n3=2 1/C
p
n
and thus that
2
3
n3=2C
1
3
S
2
3
n3=2C
p
n
2
3
:
In other words, the sum is very close to^23 n3=2.
We’ll be using Theorem 14.3.2 extensively going forward. At the end of this
chapter, we will also introduce some notation that expresses phrases like “the sum
is very close to” in a more precise mathematical manner. But first, we’ll see how
Theorem 14.3.2 can be used to resolve a classic paradox in structural engineering.
14.4 Hanging Out Over the Edge
Suppose we havenidentical unit length rectangular blocks that are uniformly
weighted. We want to stack them one on top of the next on a table as shown in
Figure 14.4. Is there some value ofnfor which it is possible to arrange the stack
so that one of the blocks hangs out completely over the edge of the table without
having the stack fall over? (You are not allowed to use glue or otherwise hold the
stack in position.)
Most people’s first response to this question —sometimes also their second and
third responses —is “No. No block will ever get completely past the edge of the
table.” But in fact, ifnis large enough, you can get the top block to stick out as far
as you want: one block-length, two block-lengths, any number of block-lengths!
14.4.1 Stability
A stack of blocks is said to bestableif it will not fall over of its own accord. For
example, the stack illustrated in Figure 14.4 is not stable because the top block is
sure to fall over. This is because the center or mass of the top block is hanging out
over air.
In general, a stack ofnblocks will be stable if and only if the center of mass of
the topiblocks sits over the.iC1/st block foriD 1 , 2,... ,n 1 , and over the
table foriDn.
We define theoverhangof a stable stack to be the distance between the edge of
the table and the rightmost end of the rightmost block in the stack. Our goal is thus
to maximize the overhang of a stable stack.
