Sqrt(2) ^^ x is approximated better and better by 2 - a*ln(2)^x + (a^2/(4*(1 - 1/ln(2))))*ln(2)^(2*x), where a is lim x → ∞ (2 - sqrt(2) ^^ x) / ln(2) ^ x. It has an error term of big O of ln(2) ^ 3x. Does anyone know a formula for the asymptotic approximations of a ^^ x when 1 < a < η? Or even more terms, but not a formula for them?

All of the higher order terms for b^^x are expressible, in closed form in terms of b's version of a. For base b the analog of a is lim x → ∞ (LambertW(-ln(b))/ln(b)-b^^x)/LambertW(-ln(a))^b.

All of the higher order terms for b^^x are expressible, in closed form in terms of b's version of a. For base b the analog of a is lim x → ∞ (LambertW(-ln(b))/ln(b)-b^^x)/LambertW(-ln(a))^b.

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ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\

ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\