I am hoping to obtain a closed-form solution or an asymptotic result for the recurrence relation

$$y_{n+1}=y_n+a+\frac{b}{y_n}$$

Any help would be very much appreciated!

  • Do you have any indication that such a closed form exists, or just hoping? – AlexanderJ93 Sep 13 at 18:40
  • 1
    I have no indication that a closed-form expression is possible, but an asymptotic result in the limit of large n seems likely. A plot seems to suggest that the difference y[n+1]-y[n] decays hyperbolically in n. – Alex Sep 13 at 18:58
up vote 3 down vote accepted

I'm going to assume that $a, b, y_0$ are all positive.

From $y_{n+1} =y_n+a+\frac{b}{y_n+a} $ it follows that $y_{n+1}-y_n =a+\frac{b}{y_n+a} $ so that $y_n \gt na+y_0 $. Therefore $y_{n+1}-y_n \lt a+\frac{b}{(n+1)a+y_0} $.

Summing, $y_{n}-y_0 \lt na+b\sum_{k=0}^{n-1} \frac1{(n+1)a+y_0} \lt na+\frac{b}{a}(\ln(n)+c) $ so $y_n \lt na+\frac{b}{a}\ln(n)+d $ where $d$ is a computable constant.

From this lower bound, we get

$\begin{array}\\ y_{n+1}-y_n &\gt a+\frac{b}{na+\frac{b}{a}\ln(n)+d+a}\\ &= a+\frac{b}{(n+1)a+(b/a)\ln(n)+d}\\ &= a+\frac{b}{(n+1)a}\frac1{1+\frac{(b/a)\ln(n)+d}{(n+1)a}}\\ &= a+\frac{b}{(n+1)a}\frac1{1+\frac{b\ln(n)+da}{(n+1)a^2}}\\ &> a+\frac{b}{(n+1)a}(1-\frac{b\ln(n)+da}{(n+1)a^2}) \quad\text{since }\frac1{1+x} > 1-x\\ &= a+\frac{b}{(n+1)a}-\frac{b(b\ln(n)+da)}{(n+1)^2a^3}\\ \end{array} $

Since $\sum \frac{\ln(n)}{n^2}$ converges, summing this gives $y_n \gt na+\frac{b}{a}\ln(n) + C$ for come constant $C$.

More effort might yield $\lim_{n \to \infty} y_n- na-\frac{b}{a}\ln(n) $ existing (and it probably does), but I'll stop here.

Note: This is a fairly standard upper-lower bound technique for getting asymptotics.

  • 1
    Thank you so much for this nice write-up. I really appreciate it. – Alex Sep 13 at 19:40

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