how-to-calculate-lim-n-0-05-0-05-0-05-0-05-n-in-a-simple-and-fast-way-

Question Number 61313 by Tony Lin last updated on 31/May/19

h o w t o c a l c u l a t e \underset{n \to + \infty}{\lim 0} . 05^{0 . 05^{0 . 05^{.^{.^{. 0.05}}}}}} n

i n a s i m p l e a n d f a s t w a y ?

Commented by mr W last updated on 31/May/19

l e t x = 0 . 05^{0 . 05^{0 . 05^{.^{.^{. 0.05}}}}}} n \to \infty

\Rightarrow x = 0 . 05^{x}

\Rightarrow x = e^{x \ln 0.05}

\Rightarrow {x e}^{- x \ln 0.05} = 1

\Rightarrow (- x \ln 0.05) e^{- x \ln 0.05} = - \ln 0.05

\Rightarrow - x \ln 0.05 = W (- \ln 0.05)

\Rightarrow x = \frac{W (- \ln 0.05)}{(- \ln 0.05)} = \frac{1.0492}{2.9957} = 0.3502

i . e . \underset{n \to + \infty}{\lim 0} . 05^{0 . 05^{0 . 05^{.^{.^{. 0.05}}}}}} n \approx 0.3502

Commented by mr W last updated on 01/Jun/19

g e n e r a l l y f o r a < 1 :

a^{a^{a^{\dots . .}}} = \frac{W (- \ln a)}{(- \ln a)}

f o r a = 1 :

a^{a^{a^{\dots . .}}} = 1

f o r a > 1 :

a^{a^{a^{\dots . .}}} \to \infty

Commented by Tony Lin last updated on 01/Jun/19

t h a n k s s i r, i t i s c o o l!