# Convergence of a Sequence

1. Oct 15, 2005

### Icebreaker

"Let $$k\in \mathbb{N}$$ and $$a_0=k$$. Let $$a_n=\sqrt{k+a_{n-1}}, \forall n\geq1$$ Prove that $$a_n$$ converges."

If we look at the similar sequence b_0 = k and b_n = sqrt(a_n-1), then that sequence obviously converges to 1. Unfortunately, b_n<a_n so I can't use the squeeze theorem.

Any hints would be nice.

2. Oct 15, 2005

### 1800bigk

I would say let lim an = s also lim an-1 would still be s so you can use the limit properties and can get a quadratic with s^2 - s -k=0 you should be able to go from there

3. Oct 16, 2005

### devious_

But you don't know if $\lim a_n$ exists.

Have you tried checking if a_n is monotonic & bounded?

4. Oct 16, 2005

### siddharth

You should first try to prove that the sequence is bounded.
Then if you show that it monotonically increases or decreases, you can prove that the sequence is convergent.

5. Oct 16, 2005

### Icebreaker

It can easily be shown that it's monotonically increasing. However, it's the bounded part that gets me. Maybe I can use Herschfeld's Convergence Theorem?

Last edited by a moderator: Oct 16, 2005
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