If, x x x . . . = 2 , then what is x?(that is x^x^x and so on)
Answer up to two decimal places.
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This is not complete. By your logic, x x x . . . = 4 has a solution of x = 2 as well.
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We define the recursive sequence
{ a 0 = 2 a n + 1 = 2 a n , n ∈ N
We can show inductively that a n is increasing and bounded above by 2 , implying that a n is indeed convergent. Let a n → x . Clearly, 2 < x ≤ 2 . But a n + 1 = 2 a n → x , as well.
Hence x = n → ∞ lim a n = n → ∞ lim a n + 1 = n → ∞ lim 2 a n = 2 x . Thus x satisfies 2 x = x . Finally, we show that 2 x > x ∀ x ∈ ( 2 , 2 ) . Thus, we have x = 2 .
Yeah, i think you first have to prove that the power tower converges.
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x x x . . . = 2
ln x x x . . . = ln 2
x x x . . . ln x = ln 2
2 ln x = ln 2
x = 2 1 ln 2 = ln 2
x = 2