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Let $a_1=1$, $a_n=\sqrt{n+a_{n-1}}$, $n\geq 1$. Show that $a_n=\sqrt{n}+\frac{1}{2}+\frac{1}{8\sqrt{n}}+o(\frac{1}{\sqrt{n}})$.

How to prove, and is there any general method?

xldd
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1 Answers1

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Hints: First, prove $a_n<\sqrt{2n}$ for all $n$ (or anything similar in nature). Second, use the binomial theorem on $$\sqrt{n+a_{n-1}}=\sqrt{n}\cdot\left(1+\frac{a_{n-1}}{n}\right)^{1/2}.$$ The first part guarantees that the binomial expansion works since $a_{n-1}/n<1$.

Can you handle the rest?

Clayton
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  • It is a little bit hard. I have seen $1/2$, $-1/8$ etc. But I could not go further. – xldd Jul 26 '19 at 05:48
  • @xldd: Indeed, the trick is to use this type of asymptotic twice: once for $a_n$ then again for $a_{n-1}$. – Clayton Jul 26 '19 at 13:06