TITLE: What is $2!!!!!!!!!!!!!!!!!!!!$................. (up to?
QUESTION [6 upvotes]: A few days back a question came to my mind

What is the value of $2!!!!!!!!!!!!!!!!....$ (up to infinity)?

I feel it is 2, but one of my friends said that we can't say that for infinity.
I know it comes out to be 2 for any finite value.
But what about infinity?

REPLY [2 votes]: Just write $a_n:=2!\cdots!$ with $n$ exclamation marks. You want to compute $\lim_{n\to+\infty}a_n$. But you can easily show by induction that $a_n=2\;\forall n\in\Bbb N$, hence you're searching the limit of a constant sequence.