Inspired by Abhay Tiwari
x x ⋅ ⋅ x e = 2
Find all positive solutions x of the equation above.
Clarification : The LHS represents an infinite power tower.
Inspiration and still more inspiration .
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1 solution
Would that still work if we had a 4 or a 5 on top of the tower, instead of an e ?
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Of course! Because we have an infinite number of 'x' the question is what comes after infinity? So I guess the most above exponent doesn't play any role!
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Oh but it does. Just plug in 4 on top and work your way down, step by step.
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@Otto Bretscher – Behavior is very curious then. Numeric shows that for x = s q r t 2 the tower is infinity. However for not too small deviations from x it has almost value 2. With an e on top x is exact s q r t 2 .
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@Andreas Wendler – What you want to explore is the situation where x = 2 and we vary the number on top.
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@Otto Bretscher – For natural numbers on the top up to '8' I get:
0, 1, 2, 3, 6, 8: tower converges against 2
4, 5, 7: tower diverges
Since 2<e<3 the tower therefore converges for x = 2 !
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@Andreas Wendler – It would be interesting to see the graph of this result and when/where it starts to diverge... @Otto Bretscher @Abhay Tiwari
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@Geoff Pilling – Sir, what I observed was that calculating it through Lambert
a = − l n ( e 2 ) W ( − l n ( e 2 ) ) ≈ 2 , it's an exception that there was an e at the top, This equation will not hold for any value at the top other than 2 < x < 3 .
Sir, I recently solved a question posted by Abhay Tiwari named Inspired by Otto Bretscher, Sir Chew-Seong Cheong has posted a solution for the problem. I wonder if the same method can be used to solve this question also. I am very curious about these types of prooblems as suddenly fom no where the concept of divergence and convergence jumps in. Sir, could help me a little by explaing wht it means and how to decide whether a series is converging or diverging?
Respected sir, try posting a solution to this problem Disturbing coefficients .
Can this be put as , X^2=2 ?
From power tower it follows l n x l n 2 = 2 and therefore x = 2 2 1 .