A Cuter Way to Undo Square Roots

We prove an adorable sequence of nested square roots converges to 2 using the monotone convergence theorem and some basic algebra. We are looking specifically at a recursive sequence and will find proof by induction very useful. #calculus2 #mathematics

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[fullfungo]

Here is another way to solve it.
Notice that
√a = a^(1/2)

Then we can simplify the sequence into:
√2 -> 2^(1/2),
√(2•√2) -> √2•√√2 -> 2^(1/2+1/4),
√(2•√(2•√2)) -> … -> 2^(1/2+1/4+1/8),
and so on.

In general we get:
2^(1/2+1/4+1/8+…) as the terms.

Now to take the limit we notice that 2^x is a continuous function, therefore we can bring the limit to the exponent.
lim 2^(1/2+1/4+…) = 2^(lim 1/2+1/4+…)

lim 1/2+1/4+1/8+… is well known and even has easy visual proofs. It equals 1. So the answer is
2^1 = 2.

[Unofficialtiles]

x = √(2x)
square
x² = 2x
divide by x
x = 2
simple!