Why is 1^infinity indeterminate form and not just 1? Isn’t one to any power going to be 1?
In: Mathematics
If it’s really 1 then yes, but this is when talking about limits, i.e. the 1 and infinity are are both limits of infinite sequences (or functions). In this case 1^infinity is shorthand for the limit of a(n)^b(n) where a(n) and b(n) are both infinite sequences, whose limits are 1 and infinity respectively.
For example, take a(n) = (1+1/n) and b(n) = n. In this example, the limits of a(n) and b(n) are 1 and infinity, but the limit of a(n)^b(n) = (1+1/n)^n is e (euler’s constant). On the other hand if b(n)=2n then the limit of a(n)^b(n) is e^(2).
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