- #1
twoflower
- 368
- 0
Hi all,
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.
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.