Growth Rates and the Marvelous Geometric Sequence

Students are often dazzled by the prodigious growth rate of the geometric sequence g^sub n^=2^sup n^ and the geometric series. Teachers sometimes note that the geometric sequence is the discrete form of an exponential function, which is characterized by very rapid growth. In particular, exponential functions grow faster than polynomial functions. A rigorous explanation of this claim is left to the calculus class in which students examine the relative growth rates of functions by using L'Hopital's rule. Here, Burke and Hodgson discuss the proof that g^sub n^=2^sup n^ grows faster than any polynomial sequence. They also show that the "grows faster than" relation is transitive.