TITLE: Is there a "closed" form of sequence $u_{n+1}=\frac{u_n^2}{u_n+1}$
QUESTION [6 upvotes]: Let $u_0=1$ and $u_{n+1}=\frac{u_n^2}{u_n+1}, \forall n\in \mathbb{N}$.
a) Find the formula of $u_n$?
b) Calculate the limit $\displaystyle\varlimsup_{n\rightarrow \infty} (u_n)^{\frac{1}{n}}$.

REPLY [8 votes]: Taking Didier's comment, write in fact $z_n = (1/u_n)+(1/2)$ to get recurrence
$$
z_{n+1} = z_n^2+\frac{1}{4}
$$
and then consult the literature about the Mandelbrot set.  It is known that the recurrence $z_{n+1} = z_n^2+c$ has "closed form" (in a certain precise sense$^1$) if and only if $c=0$ or $c=-2$.  Thus, in this case $c=1/4$ it has no "closed form".  
$^1$Eremenko, page 663, in: L. Rubel, "Some research problems about algebraic differential equations, II" Ill. J. Math. 36 (1992) 659--680.