Chapter 8.4, Problem 58E

Generating functions are useful in studying the number of different types of partitions of an integer n. A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the order of the integers in the sum does not matter. For example, the partitions of 5 (with no restrictions) are $1+1+1+1+1,1+1+1+2,1+1+3,$

$1+2+2,1+4,2+3$ , and 5. Exercises 53-58 illustrate some of these uses.

(Requires calculus) Use the generating function of p(n) to show that $p\left(n\right)\le e^{c\sqrt{n}}$for some constant C. [Hardy and Ramanujan showed that $p\left(n\right)\sim e^{x\sqrt{2/3}\sqrt{n}}/\left(4\sqrt{3}n\right)$, which means that the ratio of p(n) and the right-hand side approaches 1 as n approaches infinity)