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### $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$

(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 ...
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### Nested Square Roots $5^0+\sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt\dots}}}$

How would one go about computing the value of $X$, where $X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$ I have tried the standard way of squaring then trying ...
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### Nested Radicals: $\sqrt{a+\sqrt{2a+\sqrt{3a+\ldots}}}$

Let $a>0$ . How we can find the limit of : $$\sqrt{a+\sqrt{2a+\sqrt{3a+\ldots}}}$$ Thanks in advance for your help
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### Why is $\sqrt{2\sqrt{2\sqrt{2\cdots}}} = 2$? [duplicate]

Why is $\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\cdots}}}}}}$ equal to 2? Does this work for other numbers?
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### Request for example of a value of limit of a sequence where sequence does not converge

In Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$ Timothy Wagner gave a correct answer that was questioned for not having shown that the limit exists in the first place. My question ...
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### How to compute the following formulas? [duplicate]

$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ $\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$ Why they are different?
Alright, so I have a question on a little open-book challenge-test thingy that deals with repeating square roots, in a form as follows... $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$ Repeated 2012 times (...
### Proof by induction: Let $a_0=3$ and $a_{n+1}=\sqrt{a_n+7}$ if $n>0$, Prove: $3<a_n<4$
Let $a_0=3$ and $a_{n+1}=\sqrt{a_n+7}$ if $n>0$ Prove: $3<a_n<4$ At first I was quite surprised it's actually true for the base cases: $n=0$, $a_1=\sqrt{3+7}=\sqrt{10}$ $n=1$, \$a_2=...