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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?

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    $\begingroup$ Well, the finite product $\prod\limits_{n=1}^k n$ is just $k!$ ($k$ factorial). The infinite product clearly diverges, though. (However, the zeta-regularized product $\overset \infty{\hat{\prod\limits_{n=1}}}n$ is $\sqrt{2\pi}$.) $\endgroup$ – Akiva Weinberger Feb 4 '16 at 2:22
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    $\begingroup$ I would say that the answer to the question is "factorial". We have named the harmonic series even though it diverges because it is an important series for comparisons to show other series also diverge. Those comparisons are between finite partial sums, not the entire series. What is useful about this series are the finite sums. The corresponding partial products are the factorials, which are named because they are so common in their own right. The harmonic series is useful because it barely diverges. But the infinite factorial is grossly divergent, so less useful. $\endgroup$ – Paul Sinclair Feb 4 '16 at 4:03
  • $\begingroup$ @PaulSinclair, you can post that as an answer if you'd like, because that may help me. And this may be called the infinite product since we are multiplying every number up to $\infty$. There "isn't a specified answer", so this actually diverges [, and yes, grossly!]. $\endgroup$ – Obinna Nwakwue Feb 4 '16 at 23:33
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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 $\infty!$

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  • $\begingroup$ Thank you, this may help the best. $\endgroup$ – Obinna Nwakwue Feb 5 '16 at 1:11
  • $\begingroup$ Also, is there a product that either converges or is barely divergent like the Harmonic series (I'm just asking)? $\endgroup$ – Obinna Nwakwue Feb 15 '16 at 22:50
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    $\begingroup$ $$\prod_{n=1}^\infty \sqrt[n]a$$ just diverges, since it is $$a^{\sum_{n=1}^\infty 1/n}$$ However, this illustrates a point about infinite products. It is generally easy to convert them to infinite sums, so they don't need or have too much development other than what they inherit from sums. $\endgroup$ – Paul Sinclair Feb 16 '16 at 0:44
  • $\begingroup$ Sorry to respond to this over 2 months late, but that is really nice. Thanks. $\endgroup$ – Obinna Nwakwue May 9 '16 at 20:38
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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

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