I'm studying Euler's paper "On transcendental progressions, or those for which the general term is not given algebraically" (marked as [E19] in The Euler Archive), and I'm trying to understand the following fragment:
I was looking for a general expression which would give all the terms of the progression
$$1 \;+\; 1\!\cdot\!2 \;+\; 1\!\cdot\!2\!\cdot\!3 \;+\; 1\!\cdot\!2\!\cdot\!3\!\cdot\!4 \;+\; etc.$$ Supposing that, if continued to infinity, this progression would ultimately behave like a geometric progression, I came upon the following expression $$\frac{1\cdot2^n}{1+n} \cdot \frac{2^{1-n}\cdot3^n}{2+n} \cdot \frac{3^{1-n}\cdot4^n}{3+n} \cdot \frac{4^{1-n}\cdot5^n}{4+n} \cdot etc.,$$ which represents the term of order $n$ of the stated progression.
My questions are:
- Why did Euler say that this progression of factorials would ultimately behave like a geometric progression when continued to infinity?
- How could it lead him from that idea to the infinite product he shows in the very next line? How is it related to the geometric progression he was talking about? I can't see the connection right away.