Given the following function (also called infinite exponential tower):
f ( a ) = a a a a a a a a . . .
Where a is a positive real number
Let a 0 be the maximum value of a so that the function has a definite value and f ( a 0 ) = b
What is a 0 + b
Submit your answer as ⌊ 1 0 0 0 0 ( a 0 + b ) ⌋
Note: ⌊ . ⌋ is the floor function
*Note: though it may be possible to solve this problem manually all the way, my solution involves a little bit of programming. If anyone has a solution that is entirely manual, please tell me in the solution section. Thank you. *
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Let f ( a ) = x . It's easy to see that x = a x
Now we find the maximum value a 0 of a so that the equation x = a x has a real solution x 0
The solution is the intersection of two graphs: y = x and y = a x
The maximum value of a is when the line y = x is tangent to the curve y = a x
The tangent line equation y = y ′ ( x − x 0 ) + y 0 ≡ y = x where ( x 0 , y 0 ) is the point of intersection
y = a x 0 ln a ( x − x 0 ) + a x 0
We know y ′ ( x 0 ) = 1 and the tangent line goes through ( 0 , 0 ) so:
a x 0 ln a = 1 x 0 − a x 0 = 0
≡ x 0 ln a = 1 x 0 − a x 0 = 0
≡ ln a 1 − a ln a 1 = 0 x 0 − a x 0 = 0
Now we use Newton's method to solve the first equation than substitute it into the second one. This is when programming comes in handy
Newton's method: x n = x n − 1 − f ′ ( x n − 1 ) f ( x n − 1 )
Now copy and paste all of these code lines into python (I'm sorry I don't know how to put a python window here):
The program gives:
a = 1.444667861
b = 2.718281802
a + b = 4.162949663
The "entirely manual" solution works like yours - once you get to the equation lo g a 1 − a lo g a 1 = 0 , just substitute t = lo g a . The equation becomes t 1 − ( e t ) t 1 = 0 . But this is just t 1 − e = 0 , and so we find a = e e 1 .
As mentioned here , Euler showed that f ( a ) converges only when e e 1 ≤ a ≤ e e
The maximum value of the given expression is b=e, when a=e^(1/e)
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I'm not too familiar with latex, so here is a written 'manual solution' :)