-1

I am wondering what is the following limit is:

$$\lim_{a \to \infty} \prod^{\infty}_{i=a} i$$

It might be infinity because it is infinity for all real $a$.

But my intuition is telling me that the limit is undefined.

It can also be one because it is essentially an empty product.

Standard rules like algebra of limits don't apply on this limit.

wythagoras
  • 25,026
  • 2
    For any $a$ the product is not a real number. It is meaningless to talk about the limit of an undefined quantity as some variable changes. If you allow transfinite arithmetic the problem becomes quite different. – vadim123 Sep 04 '15 at 18:40
  • @vadim123 Sorry, mistyped the problem. – wythagoras Sep 04 '15 at 18:42
  • 1
    @wythagoras it is strange you're drawing a distinction between a limit at infinity to an undefined limit. A limit at infinity is already undefined. We are able to use 'at infinity' suggestively on certain undefined limits. – rschwieb Sep 04 '15 at 18:42
  • @rschwieb Depends on your conventions. I think that a limit is equals infinity is quite different form the case that the limit doesn't exist, for example because right and left limits are not equal. – wythagoras Sep 04 '15 at 18:43
  • @rschwieb So what's your point? – wythagoras Sep 04 '15 at 18:46
  • 1
    If we work in the extended real numbers, where $\infty$ is a perfectly valid number, then it's clear that for any fixed positive integer $a$, the sequence of partial products, $p_a(N) = \prod_{i=a}^{N} i$, grows without bound. So for each fixed $a$, we have $\prod_{i=a}^{\infty}i = \lim_{N \to \infty} p_a(N) = \infty$. Then $\lim_{a \to \infty} \prod_{i=a}^{\infty} i = \lim_{a \to \infty} \infty = \infty$. –  Sep 04 '15 at 18:46
  • Is this what you mean $\lim_{a \to \infty} \lim_{n \to \infty} n!/a! $ ? – drewdles Sep 04 '15 at 18:48
  • @Bungo You may post that as an answer. But when working in the normal reals, is it infinity or undefined? – wythagoras Sep 04 '15 at 18:48
  • @AnantSaxena Yes, that is an other formulation of what I mean, not sure if they are equal though. – wythagoras Sep 04 '15 at 18:49
  • @wythagoras, In the "normal reals," infinity is undefined. – Marcus M Sep 04 '15 at 18:52
  • @wythagoras check: http://math.stackexchange.com/questions/390435/factorial-of-infinity – drewdles Sep 04 '15 at 18:53

1 Answers1

3

If we work in the extended real numbers, where $\infty$ is a perfectly valid number, then it's clear that for any fixed positive integer $a$, the sequence of partial products, $p_a(N) = \prod_{i=a}^{N} i$, grows without bound. So for each fixed $a$, we have $$\prod_{i=a}^{\infty}i = \lim_{N \to \infty} p_a(N) = \infty$$ Therefore, $$\lim_{a \to \infty} \prod_{i=a}^{\infty} i = \lim_{a \to \infty} \infty = \infty$$ In the extended real numbers, this is legitimate convergence to $\infty$.

If we work in the real numbers, we still sometimes make a distinction between an arbitrary divergent sequence and one which "diverges to $\infty$", meaning that given any positive $B$, all but finitely many of the terms of the sequence exceed $B$.

In your example, working in the real numbers, we could say that for a fixed $a$, the sequence $p_a(N)$ diverges to $\infty$ as $N \to \infty$. But $\prod_{i=a}^{\infty}i$ is not a real number, so the expression $\lim_{a \to \infty}\prod_{i=a}^{\infty} i$ does not make any sense in $\mathbb{R}$. You can't talk about a limit of a sequence where the members of the sequence are not elements of the space in which you are working. For this reason, as @rschweib points out, you need to work in the extended reals for the question to make sense.

  • Good answer. In reflection, there's no choice but to work in the extended real line because otherwise we would be looking at a sequence of undefined quantities. So it's the most natural environment for this problem. – rschwieb Sep 04 '15 at 19:03
  • @rschwieb: It's a good point that in the reals, $\prod_{i=a}^{\infty}i$ is undefined, so the expression $\lim_{a \to \infty} \prod_{i=a}^{\infty}$ doesn't even make sense. I'm going to edit to mention this explicitly. –  Sep 04 '15 at 19:11