Nested Radicals and Infinite Fractions

What is $\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}}$, with the pattern continuing forever? There's no bottom layer to start computing from — every attempt to evaluate it from the inside out just finds another identical square root waiting underneath. The trick is to stop trying to compute it layer by layer and instead notice that the whole infinite expression looks exactly like itself, one layer in.

The idea: self-similarity gives you an equation

If an infinite expression $x$ contains a copy of itself nested one level down, you can replace that inner copy with $x$ itself and solve the resulting finite equation. This works because the expression is defined as a limit of a repeating pattern — assuming that limit exists and is well-behaved, the value one layer down is the same value the whole tower converges to.

For the nested radical above, let $x = \sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$. Everything under the first square root is again $2$ plus the same infinite radical, so:

$$x = \sqrt{2+x} .$$

In plain English: peeling off just the outermost square root leaves exactly the same infinite pattern underneath, so the whole thing satisfies an equation relating it to itself — an infinite object reduced to solving one algebraic equation.

Worked example 1: solving the nested radical

Square both sides of $x=\sqrt{2+x}$: $x^2 = 2+x$, i.e. $x^2 - x - 2 = 0$, which factors as $(x-2)(x+1)=0$, giving $x=2$ or $x=-1$. Since a square root is never negative, discard $x=-1$: the infinite nested radical equals exactly $2$. Sanity check by computing a few layers by hand: $\sqrt{2}\approx1.41$, $\sqrt{2+1.41}\approx1.85$, $\sqrt{2+1.85}\approx1.96$ — climbing steadily toward $2$, consistent with the algebraic answer.

Worked example 2: the golden ratio as an infinite fraction

Evaluate $x = 1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}$. The pattern one layer down is again the same infinite fraction, so $x = 1 + \dfrac{1}{x}$. Multiplying through by $x$: $x^2 = x+1$, i.e. $x^2-x-1=0$, which by the quadratic formula gives $x = \dfrac{1+\sqrt5}{2} \approx 1.618$ — the golden ratio, discarding the negative root since the fraction is built from positive terms. This is the same self-similarity trick applied to a continued fraction instead of a nested radical, and it's the standard way the golden ratio shows up as "the number that equals one plus its own reciprocal."