Name of $\prod_{n = 1}^{\infty}n = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots$

Question

I already know about the Harmonic series: $$\sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \cdot \cdot \cdot$$ But is there a name for this infinite product series: $$\prod_{n = 1}^{\infty}n = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots$$ It is not that I need any help, but I just want to know... what is the name for that above infinite product series?

Answer 1

An expanded version of my comment:

The harmonic series has a name because it is highly useful. This utility comes from the fact that it is just barely divergent. A great many sequences that are also close to converging can be seen to diverge because of comparisons to the harmonic series. This includes not just series that majorize the harmonic series, but also some that are marjorized by it, but fall to more subtle comparisons. Since this is something that occurs regularly, it is easier to talk about when the comparison series has a name. Thus the name "harmonic" series became standard. (It may have been given for other reasons, but I would argue that this is the common reason for people to use the name.)

However, $\prod_{n=1}^\infty n$ is practically never useful. Unlike the harmonic series, this product is far away from convergent, and comparable products are obviously divergent as well, so there is little need to involve it.

Since the finite subproducts are the factorials, the most sensible name to me for this product would be the "infinite factorial", or $\infty!$

Answer 2

You can call it Euler's nugget product , using the analogy from [Euler's Nugget]. 1

Dealing with divergent series or products is not as useless a concept as some might suggest, the divergent series and products do have uses e.g. see this