How does $1^\infty=\infty$?

Question

I remember hearing in school long ago that $1^\infty=\infty$.

I was just wondering if anyone could explain this in laymen's terms?

Answer 1

$1^\infty$ is indeterminate, as this Wikipedia entry explains. Or perhaps you remember something like $\lim_{n\to\infty}\sqrt[n]n=1$ ?

Answer 2

It depends on what you mean by "$1^\infty$". If you mean $\lim_{\,n\rightarrow\infty\!}1^n,$ then it can be taken to mean $1$. But generally it is not something that a mathematician would write. The most sensible thing to do is to regard arithmetic operations involving the symbol $\infty$ as meaningless. You can play around with such symbols in a very limited way, once you have understood some basic maths, but the symbol probably causes more confusion than enlightenment, and there is a good case for saying that we would be better off without it, except as an arbitrary name in the one-point compactification of the complex numbers (or, in the dual form $\pm\infty$, in the two-point compactification of the real numbers). If you don't understand this or the other comments or answers, it's not a problem: just treat "arithmetic with infinity" as meaningless, and you will be wiser than most of your colleagues.