Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The question is:

$0\leq\alpha\leq \frac{1}{4}$, We'll define $a_1=\alpha$, $a_{n+1}=\alpha+a_n^2$. Prove $(a_n)_{n=1}^\infty$ is converging and find it's limit.

I'm really confused about it, so I tried taking the private case of $\alpha=\frac{1}{4}$ but I still don't really understand it.
$a_1=\frac{1}{4}$
$a_2=\frac{5}{16}$
$a_3=\frac{89}{256}$
And I'm not really sure where this is going to or what I'm supposed to do with it.

share|improve this question
add comment

2 Answers 2

up vote 2 down vote accepted

Well, there is a way that often works on problems like this. First, suppose that the sequence does converge and find its limit. Hint: if $(a_n) \to x$, then $x = \frac{1}{4} + x^2$.

Next, when you've found the supposed limit value $x$, you'll just need to prove that $a_n - x \to 0$, and the problem will be solved. In the actual solution you can even omit the part where you determine the value of $x$. You can just say out of the blue something like "Let's show that $(a_n)$ converges to <here you name the limit>", and then go on proving that $a_n - x \to 0$. This looks cool and makes people think that you've come up with the limit value using your incredible intuition )).

share|improve this answer
1  
Then the crucial ingredient of the proof is indeed to show that $a_n-x\to0$. Your post should include a way to prove that... –  Did Nov 28 '12 at 18:28
    
@did I mean this as a hint. When $x$ is known, proving that $a_n - x$ converges to $0$ becomes way easier than the original problem was. I believe the OP can do it himself. –  Dan Shved Nov 28 '12 at 18:29
    
Sorry but I do not share your optimism. –  Did Nov 28 '12 at 18:33
1  
@did Well, if the OP wants it, he can ask me for the rest of the proof and I'll provide it. This isn't optimism. Rather, I think that this is how homework questions should be answered. I must have read something about it in the FAQs, but I cannot find it... –  Dan Shved Nov 28 '12 at 18:40
    
Thank you! That's a nice trick i didn't know... Too bad classes here doesn't bother to teach you how to solve the exercises you get... –  Nescio Nov 28 '12 at 19:24
show 1 more comment

Prove that the sequence is increasing and bounded. Both using induction.
Then find the limit as Dan Shved proposed.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.