- #1

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I don't fully understand solving of limits when the sequence is given by some recurrent expression.

Eg. I have this sequence:

[tex]

a_{n} = \sqrt{2}

[/tex]

[tex]

a_{n+1} = \sqrt{2 + a_{n}}

[/tex]

[tex]

\lim_{n \rightarrow \infty} a_{n} = ?

[/tex]

First, I should prove the monotony and finiteness (is it ok to say it in english this way?). Well I did the proof the monotony by induction, I hope right. Now the finiteness. How should I do it? Can I just guess it won't get greater than. let's say 2 ? Ok, I chose 2 and prove that it is finite.

Now the limit. Our teacher wrote this:

[tex]

\lim_{n \rightarrow \infty} a_{n} = A

[/tex]

[tex]

A = \sqrt{2 + A}

[/tex]

And I ask, what should this mean? Where does this equality come from?

Of course to solve it is easy and we find out A = 2, which is the limit. But I don't understand why the equality.

Thank you for any help.