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When pairs of terms are equal (MRB and MKB constants)

The following may be too elementary for constructive comments, but if not, then join in!

The MRB constant = Limit[Sum[(-1)^n n^(1/n),{n,1,2N}],N->Infinity]

In the domain of terms of the MRB constant ask, when are the pairs of terms equal?

•   Limit[x^(1/x) - (x + h)^(1/(x + h)) ,h->Infinity]  == 0 when x=1 because Limit[x^(1/x),x->Infinity]=1.
•   x^(1/x) - (x + 2)^(1/(x + 2)) == 0 when x=2 because 2^(1/2)=4^(1/4).
•   I think there are no more.

The MKB constant = Limit[Integrate[(-1)^x x^(1/x),{x,1,2N}],N->Infinity].

Compare the previous list to one using the domain of terms of the MKB constant, and ask when pairs of terms are equal?

•   For x != 0, x^(1/x) - (x + 0)^(1/(x + 0)) == 0 because, for example, Limit[x^(1/x) - (x + 10^-h)^(1/(x + 10^-h)), h -> Infinity]=0; see last line a special such x.
•   x^(1/x) - (x + 1)^(1/(x + 1)) == 0 when x= 2.2931662874… (By definition Foias’ second constant. See second constant at http://mathworld.wolfram.com/FoiasConstant.html).
•   x^(1/x) - (x + 2)^(1/(x + 2)) == 0 when x=2.
•   x^(1/x) - (x + 3)^(1/(x + 3)) == 0 when x= 1.801627661…
•   x^(1/x) - (x + 4)^(1/(x + 4)) == 0 when x= 1.6647142806…
•   x^(1/x) - (x + 10)^(1/(x + 10)) == 0 when x= 1.3295905071…
•   …
•   x^(1/x) - (x + 100)^(1/(x + 100)) == 0 when x= 1.00697415301373…
•   …
•   Limit[x^(1/x) - (x + h)^(1/(x + h)) ,h->Infinity]  == 0 when x=1 because Limit[x^(1/x),x->Infinity]=1, and that is where the sequence very slowly goes to.
•   Many more.
•   x^(1/x) - (x + 10^-1)^(1/(x +10^- 1)) == 0 when x 2.669048059942…
•   x^(1/x) - (x + 10^-2)^(1/(x + 10^-2)) == 0 when x= 2.713289492595…
•   x^(1/x) - (x + 10^-3)^(1/(x + 10^-3)) == 0 when x= 2.71778190…
•   x^(1/x) - (x + 4)^(1/(x +10^- 4)) == 0 when x= 2.71823182922…
•   …
•   x^(1/x) - (x + 10^-10)^(1/(x + 10^-10)) == 0, when x= 2.718281828…
•   …
•   Many more.
•   Limit[x^(1/x) - (x +10^- h)^(1/(x + 10^-h)) ,h->Infinity]  == 0 when x=E, because that is where the sequence very rapidly goes to!
POSTED BY: Marvin Ray Burns
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