What is the value of 1 to the power of infinity?

[Ralph Spencer]

One of our senior teachers talked about infinity and said that 1^∞ is not defined. On deeper probing, he said that it is a bit higher mathematics and it wouldn't be appropriate to go deeper here. Naturally, I could think of a inductive proof that it should be 1, if ∞ ∊ N. I can't think of a reason why this is untrue.

 

[Wizlem]

While it is true that \lim_{n \rightarrow \infty} 1^n = 1. If you take log(1^\infty), you end up with \infty\cdot0 which is undefined. One issue is there are many different ways to write an undefined term. You can write 0/0 as \lim_{x \rightarrow 0} \frac{x}{x} and see it equals 1 but 0/0 could mean anything. A good example of 1^\infty is the continuous interest formula \lim_{n \rightarrow \infty} (1+\frac{r}{n})^{tn}. With knowledge of l'hopitals rule, you can evaluate this to be e^{rt}.

 

[Hurkyl]

I can't think of a reason why this is untrue.

It's not true of integer exponentiation, because +∞ is not an integer. number.

It's not true of real number exponentiation, because +∞ is not a real number.

It's not true of extended real number exponentiation, because mathematicians find it preferable to leave arithmetic undefined at points where it cannot be continuous.

Why cannot extended real number exponentiation be continuous at 1^+∞? Assume that it really is. Then,

1^{+\infty} = \lim_{x \rightarrow 0} (1 + x^2)^{+\infty} = \lim_{x \rightarrow 0} (+\infty) = +\infty

1^{+\infty} = \lim_{x \rightarrow +\infty} 1^x = \lim_{x \rightarrow +\infty} 1 = 1
​

which contradicts the fact that 1 and +∞ are not equal.

 

[darkside00]

not defined as 1 to the root anything is always going to be 1