Prove the convergence of the sequence.

Question

Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$

$$x_{n+1} = \sqrt{a + x_n}$$

Welcome to MSE! Do you have any thoughts on the problem and can share your approach? It helps the MSE provide better feedback. Regards — Amzoti, May 19 '13 at 17:21

score 2 · Answer 1 · answered May 19 '13 at 17:28

2

Hint :

Prove that the sequence $\{x_n\}$ is bounded and monotonic

answered May 19 '13 at 17:28

Dimitris

You're using a proved theorem, but how can I prove that the sequence is bounded ? – Otávio Rapôso May 19 '13 at 18:08
Can you prove by induction that $x_n<a+1,\forall n\in \mathbb N$? Then the sequence will be incresasing and bounded above, therefore convergent. – Dimitris May 19 '13 at 18:13

score 1 · Answer 2 · answered May 19 '13 at 17:28

1

(Of couse we need $a\ge 0$ as premise)

(You can facilitate things a bit by prepending a term $x_0=0$. Not that then still $x_1=\sqrt{a+x_0}$)

answered May 19 '13 at 17:28

Hagen von Eitzen

2 Answers2