A Power Tower | Problem 372

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Опубликовано:

23 сен 2024

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Комментарии : 26

@angelishify 6 часов назад

But who says the expression on the left even exists? On the left is real, on the right imaginary. Even if we take n < 0.

@CarlBach-ol9zb 3 часа назад

To be fair, i is also defined as a real to the power of a real, (-1)^0.5. That said, this does feel sus

@Qermaq 2 часа назад

i^i is well-known to be e^(-pi/2 + 2pik). Matt Parker even did a video on it. It's not really that unusual for a complex number to a complex power to be a real value.

@CarlBach-ol9zb 2 часа назад

@@Qermaq, while true, this expression feels too sus. None of the numbers ever become negative or complex. So, how can a bunch of *positive* real in a power tower result in complex numbers? Additionally, power towers stop converging for reals after e^1/e, so the left expression is simply divergent under just real arithmetic. Either way, this is an artefact of the method used, and not a valid solution.

@quneptune 7 часов назад

the problem is that power towers are solved from the top down, so the last value dramatically changes the final value. also z should equal z^z but when we clearly see that (e^2π)^(e^2π) does not equal e^2π i think stumbled across a new indeterminate form of ᪲z (tetration) or z^^∞

@TheLeviathan1293 2 часа назад

8:50 So yeah, that wouldn't be a solution, since the tower goes to infinity.

@stevestarcke 7 часов назад

I tried to solve it by assuming the answer remained the same by repeated exponentiation. Playing with my slide rule (look, I'm just an old engineer) I was quickly driven to 1 as a solution. And e to the pi/2 just blows up if I keep exponentiating it. What am I missing? Love your videos!

@andrei_nazimov 8 часов назад

e^(π/2)=4,81~ 4,81^4,81^4,81^...=i ???

@klofat 4 часа назад

4,81^4,81^4,81^...=i is true if you really never stop. Like 1+2+3+... = -1/12 IF you NEVER stop. Human mind isn´t made for infinities.

@CarlBach-ol9zb 2 часа назад

@@klofat, wrong, the latter is plain wrong. It's a common misconception, the sum of natural numbers doesn't give that. This misconception stemmed from an extension (analytic continuation) for the Riemann Zeta function, which is defined for inputs with negative real parts, being portrayed incorrectly. What this video shows, though, idk if it's valid. But since I don't know enough to validate, I will merely say it's sus instead of pretending to be a wannabe know-it-all dumbass.

@CarlBach-ol9zb 2 часа назад

Eh. I don't think this might be valid. Don't power towers start diverging after some value? IIRC it was e^1/e for real exponents. For imaginary or complex results, this might not even be valid as far as I know. More proofs or work needed IMO.

@scottleung9587 7 часов назад

Got it!

@Qermaq 2 часа назад

Real values in a power tower only converge if the value is positive but below e^(1/e) (I'm not certain of the lower bound). You did a video on how sqrt(2) in a power tower seems to equal both 2 and 4, but 2 is the actual value. Maybe something like that's going on?

@0over0 3 часа назад

I just discovered something AMAZING, for real values z. If z = 1.4, then z^z^z ... CONVERGES to 1.8866....! But for z = 1.5, it diverges. There's a threshhold between 1.4 and 1.5. Q1: What is the threshold value? Q2: how quickly does IT converge?

@Random_username_1234 3 часа назад

The value is e^(1/e)

@erichendriks2807 2 часа назад

z^i=i=exp(PI i/2=> z=exp(PI/2) This is a real number, so even if the tower would converge it cannot be a solution of the equation. But the tower diverges. I think this is bad math.

@lawrencejelsma8118 7 часов назад

I think you did an infinity play to get an incorrect relationship such as 0/0 errors. When you got Z = e^[(iπ/2) + 2nπ] it is then e^i(π/2)e^i(2nπ) ... But e^(i2πn) = 1 for all integers "n" leaving: Z = e^(iπ/2) instead of your final answer that proves i = 1 incorrect result by infinite misused mathematics ... Or √-1 = √+1 ... or -1 = +1 similar error of a 0/0 problem.

@lawrencejelsma8118 7 часов назад

At 7:38 you have e^i(π/2 + 2nπ) that equals strictly the principle e^i(π/2) = i part in your math. You will never get e^(iZ) = the right expression if e^(Z) = the same right expression somehow in your mathematics! 🙄🤨🤯