September 15 Information

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) reply

One of the classic ways of defining e is

${\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$

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) reply

In general:

${\displaystyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$

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) reply

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) reply

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

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

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

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

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

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

@ Georgia guy: 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) reply

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) reply