5.5 INEQUALITIES 175In summary, the hierarchy of growth rates, from slowest to highest, islogarithms, powers, exponents.
Simple Proofs
Of the many ways of proving inequalities, the simplest is to perform operations that
create logically equivalent but simpler inequalities. More sophisticated variants in
clude a little massage, as well. Here are some examples.
Example 5.5.9 Which is larger, V19 + V99, or JW + V98?
Solution: We shall use the convention of writing a question mark (?) for an
alleged inequality. Then we can keep track. If the algebra preserves the direction of
the alleged inequality, we keep using the question mark. If instead we do something
that reverses the direction of the alleged inequality (for example, taking reciprocals of
both sides), we change the question mark to an upside-down question mark (,,). So we
start withV19+V^99? VW+V^98.
Squaring both sides yields19+2)19·99 +99? 20+2V20 ·98+98,
which reduces toThis of course is equivalent to(^19) ·99? 20· (^98).
At this point we can just do the calculation, but let's use our factoring skills: Subtract
19. 98 from both sides to get
19· 99 - 19· 98? 98,
which reduces to19? 98.
Finally, we can replace the "?" with "<" and we conclude that
V19+V (^99) < VW+V (^98).
E 5 5 10 h· h· b·
1998 1999?
xample •. W IC IS Igger,1999
or2000
.
Solution: This can be done in many ways; here is an argument that uses the define
a function tool. Let