'Prove' 4 = 2. Can You Spot The Mistake?

Many people emailed me this apparent paradox showing 4 = 2. Can you figure out where the mistake is?

My blog post for this video

Award winning paper Knoebel, R. Arthur. "Exponentials reiterated." The American Mathematical Monthly 88.4 (1981): 235-252.

Wikipedia tetration

False proof 2 equals 4

Math StackExchange links

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[MindYourDecisions]

Links from video description:

Award winning paper Knoebel, R. Arthur. "Exponentials reiterated." The American Mathematical Monthly 88.4 (1981): 235-252.

Wikipedia tetration

False proof 2 equals 4

Math StackExchange links

[Brooke-rwrc]

Person: "2 = 4!"
Physicist: "That makes sense, they're both basically 3."

[GourangaPL]

Ok, so, infinite number of mathematicians get into a bar. First orders a beer. Second orders 1/2 beer, 3rd orders 1/4 beer, 4th orders 1/8 beer, barman gives them two beers and says "guys, you have to know your limits" :)

[amanahtigetige]

5x0=0
100.000.000.000x0=0
Wow 5 and 100.000.000.000 are *same*

[yourlocalbeeswarm]

the error is the fact that 4 isn't the same as two

[chinareds]

"Whenever you have an infinite number of items, you generally need to check for convergence before you simply start doing substitutions and simple arithmetic to get an answer" .... unless your name is Ramanujan.

[TheDannytaz]

I remember when I first showed the initial problem to one of our lecturers, he said: "This safely assumes there is a solution yes?" Which we replied yeah. We then showed him, excitedly, the solution but he replied 'The real question here is why does it work for 2?' I personally shrugged it out, but years later I see what he was trying to tell us.

[andygong]

“1=0!”
Math teacher: “well that makes sense...”

[caiogaiotto]

When we mess with infinity, things get a little awkward

[JohnLeePettimoreIII]

I don't know about anyone else, but my mistake was hitting the play button at this time of night.

[problematicpuzzlechannel]

Nice problem. Always have to be careful with convergence!!

[felissilvestris]

Nobody:

Engineers:
Pi=3=e
chANgE mY mInDE

[vladimbond]

6:08 "whenever you have an infinite number of items, ..." Yeah sure all the time

[hasangarmarudi]

If you equate x^x^x^...= y and solve for x, you will get that x = y^(1/y) which you can write as the function f(x) = x^(1/x) and find that this function is the flipped version of x^x^x^x....=y, flipped over the x=y line. This mins the maximum of the new function is the maximum range of convergence for the x iteration. And that is as said in the video, equal to e^(1/e)

[davidellis]

The sequence of finite power towers with x = sqrt(2) starts at sqrt(2) and increases monotonically, converging to 2, not 4. The sequence with x = 1.5 diverges, as does the sequence with x=1.45. As Presh points out in his video, no power tower sequence starting at any value of x converges to a value larger than e.

[maxinator]

And the best part: the curve is actually the inverse of y=x^(1/x) (with the given range and domain).

[billy.]

This video is much better than other recent ones. I learned something new.

[tkzhwkh]

I was waiting for the explanation of how you find that convergence range

[MrBrain]

Excellent video and explanation! This reminds me a lot of Mathologer's video, where he explains the problem with other channels' assertions that 1+2+3+4+5+… = -1/12.

[kouverbingham]

"can you spot the mistake? neither can we! but it's buried deep in this paper!"