A proof that 2 = 4?

Question

Let's solve the equation x^x^x^x^... = 2, where x^x^x^x^... has infinite exponents. To solve it, the x^x^x^x^... in the exponent has to equal 2 as well, so we get x^2 = 2, or x = √2.

Using the same technique for the equation x^x^x^x^... = 4, we get x^4 = 4, and x = 4^(1/4) = √2. However, we also showed that x^x^x^x... = 2, therefore 2 = 4, right??

Hmm, Scythian wants to muddy the waters...

Actually, I think Agent B gives a logical answer. If you use a calculator on √2^√2^√2^..., you'll see it does appear to be converging to 2, so it seems the power tower really does have a limit. As Agent B says, I think the reason you also get √2 as the answer when you solve x^x^x^... = 4 is because the technique is invalid in this case, the reason being 4 is outside the range where the power tower converges. It's similar to how you can get illogical results by taking logs of negative numbers.

In contrast, the limit as x → 0 of f(x) = x^x^x^... behaves differently, flip-floping between 0 and 1, so it doesn't meet the definition of a limit. Now, if you want to define something called a "bifurcated convergence", I guess this function applies. :)

Answer 1

Hey there!

Not exactly. The technique is true only for x=sqrt(2).

i.e. sqrt(2)^sqrt(2)^sqrt(2)...=2.

This is also true for x=1.

i.e. 1^1^1...=1.

Infinite exponents are a great example on Lambert's W function.

Let x^x^x...=k.

Then, x=W(-ln(k))/-ln(k).

This is a limit.

Hope it helps!

Answer 2

Okay, I can't leave Agent B's answer unchallenged. He needs to read the previous postings regarding this weird function, there's more than meets the eye. See this Y!A posting:

http://answers.yahoo.com/question/index;_ylt=AmyV7DF8EANzCWGaQXj4PYPsy6IX?qid=20070823121027AAdNxEJ

The expression involving the Lambert function gives you the limiting value of the power tower only down to x = (1/e)^e, and then the power tower function gets weird, splitting into two values, one ending in 0 and the other in 1 where x = 0. In contrast, the Lambert expression continues as a single valued function until x approaches 0, and is said to be "indeteriminate" where x = 0, but it does in fact approach 0 for small x. See my earlier, but mistaken, posting, which Zanti3 corrected.

As for Zanti3's "2 = 4 conjecture", let's say that we have a function that has two different values for a particular value x, or f(x) = y1, and y2. That does not mean that y1 = y2. It just means that the function yields two different values for that particular value x. In this case of the infinite power tower, for x = √2, the value actually has both 2 and 4 as values! Absolutely bizzare but true. Anytime we have a pair of numbers a and b such that a^b = b^a, then a^(1/a) = b^(1/b) will yield both values a and b if plugged into the infinite power tower.