most recent 30 from http://mathoverflow.net 2013-05-24T03:23:37Z http://mathoverflow.net/feeds/question/88738 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88738/behavior-of-recurrence-relation David Harris 2012-02-17T17:01:38Z 2012-02-17T21:10:37Z

<p>Consider the following recurrence:</p> <p>$$ y_i = y_{i-1} + \frac{\log(p^{y_{i-1} - y_{i-2}} - \frac{\lambda}{y_{i-1}^2})}{\log p} $$ where $0 &lt; p &lt; 1$. We are only interested in values for the constant $\lambda$ such that $y_i \geq 0$ for all $i$. We start with $y_0 = 0, y_1 = 1$.</p> <p>Some experiments with different parameters show that the sequence $y_j$ approaches $y_j \approx a j + b$ for some constant $a \geq 0, b \leq 0$. It seems like you should take $\lambda = O(p)$ but I cannot prove this.</p> <p>Are there any relatively simple bounds (upper and lower) for the asymptotic behavior of $y_j$?</p>