Here $x_1>0$ is the initial condition and $x_{n+1}$ is defined by

$$x_{n+1}=\Big[\frac{1}{n}\sum_{k=1}^n x_k^2 -\frac{1}{n^2}\Big(\sum_{k=1}^n x_k\Big)^2 \Big]^{1/2}. $$ Clearly, $x_n=\lambda_n \cdot x_1$ where $\lambda_1,\lambda_2,\dots$ is a sequence of positive real numbers not depending on the initial condition $x_1$. For instance, $\lambda_1=1,\lambda_2=0$. It seems, based on empirical evidence, that $\lambda_n\sim \gamma \cdot n^\alpha$ for some constants $\gamma\approx 0.54$ and $\alpha\approx-0.35$.

Substituting $x_k$ by its asymptotic expansion in the first formula defining the sequence $(x_n)$, one gets the following, after trivial computations and rearrangements: $\alpha^2=(2\alpha+1)(\alpha+1)^2$. Let $\tau=(13+3\sqrt{33})^{1/3}$. Then we have, according to Mathematica:


Is this correct? Also, I could not find how to determine the value of $\gamma$. Could you express $\gamma$ as a solution of some equation (as I did for $\alpha$) or using a series or anything else?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.