Domain of Infinite tetration
Infinite Tetration
Description:-
Some math enthusiasts come across such type of infinitely tetrated exponentials or or in other words we call it an "infinite power tower" ... here we are going to study such a power tower
y=x^(x^(x.....))) (till infinity) we are focusing here on the domain of function y such that y is a finite value and aiming of finding such x where it satisfies this condition...
by quick guess many mathematicians say x lies in (0,1] with a logic of divergence on overunity of x but well guyz have you ever wondered putting x=1.1 or 1.2 in your calcy and hitting a formula x^x repeateadly ?!? surprisingly it WILL CONVERGE ... hmm this is creepy .. its answer can be found out by the same oldschool trick of letting y=x^y ...(1) as its infinitely tetrated...
so differentiating it we have dy = y*x^(y-1)*dx + x^y*ln(x) dy ;
=> 1 = y*x^(y-1)*(dx/dy) + x^y*ln(x)
to find the domain let dx/dy=0...
=> 1 = x^y*ln(x) ....(2)
but we have y = x^y so considering (2),
=> ln(x) = 1/y
=> x = exp(1/y)
substuting in y = x^y, we have y = exp(1/y)^y = exp(1/y * y)= e
now substuting y = e in (1) we have x=e^(1/e)
hence this is the maximum value of x for which y is finite i.e. e after that y will overflow and approach to +INFINITY vertically...
hence x must lie in (0 , e^(1/e)] or (0,1.44466786101..] wow the higher bound is greater than 1 surprisingly ;) .....
Simulate the demo to see this interesting fact>>>