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I was dealing with the following problem:

Define the sequence $x_{n+1}=\sqrt{x_n+6}$ with $x_1=1$. Find the value of $\lim_{n\to\infty}6^n(x_n-3)$.

It was easy to see that $x_n\to 3$. However, I find it hard to solve the limit $\lim_{n\to\infty}6^n(x_n-3)$. I try to use the Taylor series of $\sqrt{6+x}$ at point $x=3$, but I cannot go further. Indeed, I was also curious about whether we can find the asymptotic $$x_n\sim3+a_1A^n+a_2A^{n+1}+\cdots+a_mA^{n+m-1}+O(A^{n+m}).$$ Can someone help me? Thanks in advance.

MathLearner
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    Here is the identical question only with a different initial value: https://math.stackexchange.com/q/4097866/42969 – Martin R Nov 07 '23 at 15:53
  • [https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem] (Stolz–Cesàro theorem) might be helpful – pie Nov 07 '23 at 16:38

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