What is the generating series of the following sequence? 2ax + 2ax2 + ax3 +…

A generating series packages an infinite sequence of coefficients into one closed-form function by matching it against a known power-series identity. The exponential series ex = 1 + x + x2/2! + x3/3! + x4/4! + x5/5! + ... is the standard identity to check whenever a sequence's terms shrink by consecutive factorials.

1. Factor 2a out of every term: 2a(x + x2 + x3/2 + x4/6 + x5/24 + x6/120 + ...).

2. Factor x out of the remaining bracket: 2ax(1 + x + x2/2 + x3/6 + x4/24 + x5/120 + ...).

3. Compare the bracket's denominators - 1, 1, 2, 6, 24, 120 - to the factorials 0!, 1!, 2!, 3!, 4!, 5!; they match exactly, so the bracket is the Maclaurin expansion of ex.

4. Substitute ex for the bracket: the generating series is 2ax times ex.

Expanding 2ax times ex back out - 2ax(1 + x + x2/2! + x3/3! + x4/4! + x5/5! + ...) - gives 2ax + 2ax2 + ax3 + ax4/3 + ax5/12 + ax6/60 + ..., which reproduces every term of the given sequence exactly, confirming the closed form independently.

The other candidate forms do not fit this factorial-decay pattern:

• a form built from 2(1 - x)n evaluates to a nonzero constant at x = 0 and carries a free parameter n that never appears in the given sequence

• a form built from 2(1 + ax)n is a finite polynomial for any fixed n, while the given sequence continues indefinitely

• a form built from 2(1 - ax)n likewise evaluates to a nonzero constant at x = 0, unlike the given sequence, which has no constant term

So the generating series of the sequence is 2ax times ex, i.e. ex · 2ax.