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September 15

Defining 1^infinity as e

Briefly, in March 2013 (study the history of the 1 article) there was a period where someone wrote that 1 to the infinity power is e. Is there any logic in this statement?? Georgia guy (talk) 21:46, 15 September 2021 (UTC)

One of the classic ways of defining e is

$e=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n$

which is an example of the indeterminate form 1^∞. So you can sort of see where it comes from. As to whether this is a better choice of a value to assign to 1^∞ than any other, I would say that it is not. --Trovatore (talk) 21:49, 15 September 2021 (UTC)

In general:

$e^x=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n.$

So by changing the value of x you can get any positive value for 1^∞. If you allow complex values for x then you can get any non-zero value. You could tweak the limit in other ways to get 1^∞ = 0 or ∞. I think it's best to leave 1^∞, along with 0/0, ∞/∞, ∞ − ∞, etc. undefined except as indeterminate forms with no assigned value. --RDBury (talk) 22:20, 15 September 2021 (UTC)

PS. "Briefly" here means for approximately 8 min. on March 11. There is also a blurb in the talk page Talk:1/Archive 1#1^∞ is e. --RDBury (talk) 22:50, 15 September 2021 (UTC)

It is indeed best left alone. One might choose to define it that way, but that makes a lot of algebra break down. You need to introduce all kinds of exceptions all over the place, or else you get things like

$e^2=(1^{\mathrm{\infty }}{)}^2=1^{2\times \mathrm{\infty }}=(1^2{)}^{\mathrm{\infty }}=1^{\mathrm{\infty }}=e.$

 --Lambiam 10:25, 18 September 2021 (UTC)

"I would quarrel with mathematics and say that the sum of many zeros is a dangerous number" (Stanislaw Jerzy Lec). And $1^{\mathrm{\infty }}$ is the exponential of it. pma 13:53, 21 September 2021 (UTC)

The sum of any number of zeros is zero. Georgia guy (talk) 14:40, 21 September 2021 (UTC)

Corollary. Zero is a dangerous number.  --Lambiam 14:48, 21 September 2021 (UTC)

Template:Re AFAIK the 'zero' in the Lec's aphorism may be a person with little knowledge (possibly also little skills) compared to their social position. --CiaPan (talk) 19:19, 21 September 2021 (UTC)

See 1 + 2 + 3 + 4 + ⋯ for some other strange formulas, but I don't remember 1^∞=e having anything like that. 67.164.113.165 (talk) 05:31, 22 September 2021 (UTC)