Identities versus bijections

Abstract

Consider the infinite product (1 + x)(1 + x ²)(1+ x ³)(1+ x ⁴) … and expand it in the usual way into a series $\sum {_{n \geqslant 0}{a_n}{x^n}}$ by grouping together those products that yield the same power x ⁿ. By inspection we find for the first terms

$$\prod\limits_{k \geqslant 1} {\left( {1 + {x^k}} \right) = 1 + x + {x^2} + 2{x^3} + 2{x^4} + 3{x^5} + 4{x^6} + 5{x^7} + ....} $$

((1))

So we have e. g. a ₆ = 4 a ₇ = 5, and we (rightfully) suspect that a _n goes to infinity with n→ ∞.