What is the following sequence classified as? I don't want to make anybody solve it, I just need to know where to begin looking to solve it.
$$\alpha_1 = \sqrt{20}$$
$$\alpha_{n+1} = \sqrt{20 + \alpha_n}$$
I am suppose to prove that it converges to 5, however if I could just get a little terminology help it is more then appreciated!
Note: I updated the terminology, as well as give the initial value.
Thanks!
Answer
First, it's not a series, it's a sequence. Fixed in the original.
Second, it's a recursively defined sequence.
A sequence is "recursively defined" if you specify some specific values and then you explain how to get the "next value" from the previous one; much like induction. Here, you are saying how to get the "next term", $\alpha_{n+1}$, if you already know the value of the $n$th term, $\alpha_n$.
Once you know the first value, then the sequence is completely determined by that first value and the "recurrence rule" $\alpha_{n+1}=\sqrt{20+\alpha_n}$.
Now some hints:
- Show the sequence is increasing.
- Show the sequence is bounded.
- Conclude the sequence converges.
- Once you know it converges, take limits on both sides of the recursion to try to figure out what it converges to.
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