<s>[INST] Series Convergence What does this series converge to? 
$$ \sqrt{3\sqrt {5\sqrt {3\sqrt {5\sqrt \cdots}}}} $$
and also this?
$$ \sqrt{6+\sqrt {6+\sqrt {6+\sqrt {6+\sqrt \cdots}}}} $$
And, generally speaking, how should one approach these kind of questions?
 [/INST] For $$\sqrt{6 + \sqrt{6 + \sqrt{6 +\dots}}}:$$
Let \begin{align*}
x &= \text{the given equation}\\
&= \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots}}}
\end{align*}
Since the series is infinite, we can write
$x = \sqrt{6 + x}$,
or  $$x^2 - x - 6 = 0.$$
Therefore, $x = 3$ or $x = -2$
Since the answer cannot be negative, reject $x = -2$. Therefore, $x = 3$.
Do the same thing for the first series.
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