<s>[INST] Limit of the nested radical $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)
For $c \gt 0$, consider the quadratic equation
$x^2 - x - c = 0, x > 0$.
Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining
$$x_{n+1} = \sqrt{c+x_n}$$
Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.
Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.
 [/INST] I am going to do this Ramanujan-style: pick some real positive $a$.
$$
a=\sqrt{a^2}=\sqrt{a^2-a+a}=\sqrt{a^2-a+\sqrt{a^2-a+a}}=\sqrt{a^2-a+\sqrt{a^2-a+\sqrt{a^2-a+a}}}=\dots=\sqrt{a^2-a+\sqrt{a^2-a+\sqrt{a^2-a+\sqrt{a^2-a+\dots}}}}
$$
What you do is keep replacing the last $a$ in the expression by $\sqrt{a^2-a+a}$.
Now let $a^2-a=c$ and we have the given expression. Solve that for $a$ and you have your answer.
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