Given the following sequence:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]x_0 = 1, \quad x_1 = \sqrt{3+1}, \quad x_2 = \sqrt{3+\sqrt{4}}, \quad x_3 = \sqrt{3+\sqrt{5}},[/tex]

[tex]x_4 = \sqrt{3+\sqrt{3+\sqrt{5}}}, \quad x_5 = \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{5}}}} \ldots[/tex]

prove the above sequence converges and determine the limit.

...................

So from [tex]n=3[/tex] onwards, I notice that the sequence is recursively defined:

[tex]n \geq 4,\;x_n = \sqrt {3 + x_{n - 1} }[/tex]

To prove convergence, I'd stimply have to show sequence is bounded above and that it's increasing.

I'm not quite sure how to do this with a recursive function.

To find the limit, I realized that

[tex]x_n = \sqrt {3 + x_{n - 1} }[/tex] is at it's "equilibrium point" when [tex]x = 3 + \sqrt{x}[/tex]. I solved for [tex]\sqrt{x}[/tex] and found the limit to be:

[tex]\frac{1\pm\sqrt{13}}{2}[/tex]. I then see that we go towards:

[tex]\frac{1+\sqrt{13}}{2}[/tex]

So I know the limit will be the above if the sequence converges.

Any help on the proof?

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# Sequence Proof

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