Wikipedia:Reference desk/Archives/Mathematics/2021 September 15

= 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
 * $$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$
 * which is an example of the indeterminate form 1&infin;. 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&infin; than any other, I would say that it is not. --Trovatore (talk) 21:49, 15 September 2021 (UTC)
 * In general:
 * $$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n.$$
 * So by changing the value of x you can get any positive value for 1&infin;. 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&infin; = 0 or &infin;. I think it's best to leave 1&infin;, along with 0/0, &infin;/&infin;, &infin; − &infin;, 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. --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
 * $$e^2=(1^\infty)^2=1^{2\times\infty}=(1^2)^\infty=1^\infty=e.$$
 * --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 $$1^\infty$$ is the exponential of it. pm a 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)
 * 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)