Question

What is `1^oo` ?

Answer 1

`1^oo = 1`

Explanation:

We are reasonably safe with this one, though if you tweak it in any way it can get messy.

What do we mean by `1^oo` ?

Perhaps you think of:

`1^n = overbrace(1 * 1 * ... * 1)^(n " times")`

So:

`1^oo = overbrace(1 * 1 * ... * 1)^(oo " times")`

That's not strictly accurate. `oo` can be thought of as a shorthand for a limiting process. So we could define:

`1^oo = lim_(n->oo) 1^n = lim_(n->oo) 1 = 1`

Footnote

To see some of the dangers in limit calculations, consider:

`lim_(n->oo) (1+1/n)^n`

We might naively put:

`lim_(n->oo) (1+1/n)^n = (1+1/oo)^oo = (1+0)^oo = 1^oo = 1" "` WRONG

Instead we need can find the true value by looking at the actual value for finite values of `n` and then taking the limit as `n->oo`.

Using the binomial theorem we actually find:

`lim_(n->oo) (1+1/n)^n = sum_(k=0)^oo 1/(k!) = e`