Convergent Sequence Terminology

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.