CHAPTER 1
Introduction
1.4
The general way to do this is to write a procedure with heading
void processFile( String fileName );
which opens fileName, does whatever processing is needed, and then closes it. If a line of the form
#include SomeFile
is detected, then the call
processFile( SomeFile );
is made recursively. Self-referential includes can be detected by keeping a list of files for which a call to
processFile has not yet terminated, and checking this list before making a new call to processFile.
1.5
public static int ones( int n )
{
if( n < 2 )
return n;
return n % 2 + ones( n / 2 );
}
1.7
(a) The proof is by induction. The theorem is clearly true for 0 < X 1, since it is true for X = 1, and for X <
1, log X is negative. It is also easy to see that the theorem holds for 1 < X 2, since it is true for X = 2, and
for X < 2, log X is at most 1. Suppose the theorem is true for p < X 2p (where p is a positive integer), and
consider any 2p < Y 4p (p 1). Then log Y = 1 + log(Y/2)< 1 + Y/2 < Y/2 + Y/2 Y, where the first
inequality follows by the inductive hypothesis.
(b) Let 2
X
= A. Then A
B
= (2
X
)
B
= 2
XB
. Thus log A
B
= XB. Since X = log A, the theorem is proved.
1.8
(a) The sum is 4/3 and follows directly from the formula.
(b)
S
1
2
3
Subtracting the first equation from the second gives
4
4
2
4
3
3S 1
1
2
4
4
2
. By part (a), 3S = 4/3 so S = 4/9.
(c)
S
1
4
9
Subtracting the first equation from the second gives
4
4
2
4
3
. 4S 1
2
3
.
4
4
2
. 4S 1
4
9
16
.
4
4
2
4
3
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