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

In summary, the conversation discusses the concept of infinity and how 1 to the power of infinity is not defined. Different ways of writing undefined terms are also mentioned, and it is noted that extended real number exponentiation is not continuous at 1+∞. The conversation concludes that 1 to the root of anything is always 1 and is undefined.
  • #1
Ralph Spencer
21
0
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.
 
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  • #2
While it is true that [tex]\lim_{n \rightarrow \infty} 1^n = 1[/tex]. If you take [tex]log(1^\infty)[/tex], you end up with [tex]\infty\cdot0[/tex] which is undefined. One issue is there are many different ways to write an undefined term. You can write 0/0 as [tex]\lim_{x \rightarrow 0} \frac{x}{x}[/tex] and see it equals 1 but 0/0 could mean anything. A good example of [tex]1^\infty[/tex] is the continuous interest formula [tex]\lim_{n \rightarrow \infty} (1+\frac{r}{n})^{tn}[/tex]. With knowledge of l'hopitals rule, you can evaluate this to be [tex]e^{rt}[/tex].
 
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  • #3
Ralph Spencer said:
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,
[tex]1^{+\infty} = \lim_{x \rightarrow 0} (1 + x^2)^{+\infty} = \lim_{x \rightarrow 0} (+\infty) = +\infty[/tex]

[tex]1^{+\infty} = \lim_{x \rightarrow +\infty} 1^x = \lim_{x \rightarrow +\infty} 1 = 1[/tex]
which contradicts the fact that 1 and +∞ are not equal.
 
  • #4
not defined as 1 to the root anything is always going to be 1
 

What is "One to the power infinity"?

"One to the power infinity" is a mathematical expression that represents raising the number 1 to an infinitely large exponent. It is typically written as 1^∞.

Is "One to the power infinity" equal to infinity?

No, "One to the power infinity" is not equal to infinity. The value of 1 raised to any positive power is still 1. However, the limit as the exponent approaches infinity is undefined.

Can "One to the power infinity" be simplified?

Yes, "One to the power infinity" can be written in terms of limits as the limit of 1^x as x approaches infinity. This limit is undefined, so the expression cannot be simplified further.

What is the value of "One to the power infinity"?

The value of "One to the power infinity" is undefined. As the exponent approaches infinity, the value of the expression becomes infinitely large, but it does not have a specific numerical value.

What is the significance of "One to the power infinity" in mathematics?

"One to the power infinity" has many applications in mathematics, particularly in calculus and limits. It is also used in the study of infinite series and sequences. Additionally, it can be used to represent certain indeterminate forms in calculus, such as 1^∞ and 0^0.

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