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I got interested in the recursion $$ a_{n+1} = a_n + \frac1{a_n} $$ in response to a question on this site (which I can no longer locate).

I thought this would be a relatively easy one to solve as an explicit function of $n$. For instance, the closely related recursion $$ b_{n+1} = \frac12 \left( b_n + \frac1{b_n} \right) $$ is the sequence of guesses in Newton's algorithm for $\sqrt{1}$, given a starting guess $b_0$, and that turns out to be $$ b_{2k} = \tanh\left( 2^{2k} x \right)\\ b_{2k+1} = \frac{1}{ \tanh\left( 2^{2k+1} x \right)} $$ with $x = \tanh^{-1} b_0$.

But the recursion for $a$ is a tougher nut to crack. Although I'd llike to have in in explicit form, that might not be practical (I tried various things, including Jacobi elliptic functions, but I nevery quite get the right identities).

This question asks to prove something about the asymptotic behavior of $a_n$ for the case of arbitrary $a_0>0$, namely that

$$ \lim_{n\to\infty} \frac{H_n^4}{a_n} = \lim_{n\to\infty} \frac{a_n} {H_n^5}=0 $$ where $H_n$ are the harmonic numbers $$H_n \equiv \sum_{m=1}^n \frac1m$$

Mark Fischler
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  • Note: it is easy to see that $a_n\nearrow\infty$, and that $a_n-a_0 = \sum_{k=1}^{n-1}\frac{1}{a_k}$. Not sure what to do from there, though. – Clement C. Jan 24 '17 at 00:02
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    Something's up with the limits in your titles: The expressions are identical. – Arthur Jan 24 '17 at 00:03
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    If we follow Jack D'Aurizio's idea and put some more effort, we can show that $$ a_n^2 = 2n + \frac{1}{2}\log n + c + \mathcal{O}\left( \frac{\log n}{n}\right) $$ for some constant $c$, where both $c$ and the implicit bound for the Big-Oh notation depend on the initial value $a_0$. – Sangchul Lee Jan 24 '17 at 00:14
  • Related: http://math.stackexchange.com/questions/29777/closed-form-for-the-sequence-defined-by-a-0-1-and-a-n1-a-n-a-n-1 – Aryabhata Jan 24 '17 at 00:26

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$a_{n+1}^2 = a_{n}^2 + 2 + \frac{1}{a_n^2}$ gives $a_n\geq \sqrt{2n-1}$ as well as $$ a_{n+1}^2\leq a_n^2+2+\frac{1}{2n-1}$$ from which $a_n\leq \sqrt{2n+O(\log n)}$.

The given limits are simple to compute given these bounds, but $$ \lim_{n\to +\infty}\frac{H_n^4}{a_n} = 0,\qquad \lim_{n\to +\infty}\frac{a_n}{H_n^5}=\color{red}{+\infty}$$ since $H_n=\log(n)+O(1)$.

Thanks to Clement C., here it is a plot of $a_n$ versus $\sqrt{2n}$:

enter image description here

Community
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Jack D'Aurizio
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    Not sure how you feel about including empirical confirmation, but if you want to include this (plot of $a_n$ vs. $\sqrt{2n}$ for $0\leq n\leq 100$) in your answer, feel free to: https://s30.postimg.org/5800t2dy9/an_vs_sqrt2n.png – Clement C. Jan 24 '17 at 00:43
  • @ClementC.: many thanks, where may I give a (+1) for the contribute? – Jack D'Aurizio Jan 24 '17 at 00:49
  • Nowhere needed -- I had made a plot, might as well not let it be wasted. – Clement C. Jan 24 '17 at 00:50
  • @Clement C. How do you get this result by evaluation \sqrt{2n}, I don't very understand, so I posted my questions here:http://math.stackexchange.com/questions/2128325/solve-the-boundary-for-the-recursion – Mclalalala Feb 05 '17 at 06:33
  • Thank you for giving me an answer – Mclalalala Feb 05 '17 at 06:34