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Let $0 < b_1 ≤ a_1$ be given. We define for $n \in \mathbb{N}$: \begin{equation} b_{n+1}:=\sqrt{a_nb_n} \\ a_{n+1}:= \dfrac{a_n+b_n}{2} \end{equation}

Are $a_n$ and $b_n$ convergent and what can be said about the limit values?

I have spent hours for this task but unfortunately I have not come up with any solution. Could anyone help me please?

Sine of the Time
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battel101
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    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Nov 14 '22 at 19:32
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    You could try googling for arithmetic-geometric mean. – Peter Phipps Nov 14 '22 at 20:38
  • "I have spent hours for this task" -- what approaches did you try (for example, application of what statements that you know about convergence and limits) in the first hour, in the second hour, and what in further hours? – Torsten Schoeneberg Nov 14 '22 at 21:04
  • Show that they are adjacent – Lelouch Nov 14 '22 at 21:34
  • @Peter Phipps Omg. I have not realised it has something to do with the arithmetic geometric mean. Obviously b_n+1 and a_n+1 have the same limit. Thank you very much – battel101 Nov 15 '22 at 10:23

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