Let $(x_n)$ be a sequence defined by $x_1=1/2, x_{n+1}=x_n-x_n^2$. It is easy to see, that $\lim\limits_{n\to\infty} x_n=0$. Now, define $y_n=n^a x_n$ for $a\in\mathbb{R}$. The question is, for which $a$ does $y_n$ converge to a nonzero limit? Obviously for $a\le0$ it converges to $0$. I think it should be $a=1$ and the limit is $1$.
It looks like the way to go is to investigate monotonicity and boundedness for various $a$. I wrote $$ y_{n+1}=\left(\frac{n+1}{n}\right)^a y_n \left(1-\frac{y_n}{n^a}\right) $$ Here I got stuck. I would appreciate any help.
Also, are the other "standard" ways to deal with such limits?
