Limit of a sequence of roots

Question

Let $L$ be a real number and $\rho \in (0, 1)$. Define the following sequence, $a_0 = 0$, $a_1 = L^\rho$, $a_i = (L + a_{i-1})^{\rho}$. Is $\lim\limits_{i \rightarrow \infty} a_i= \infty$ for all values of $L >1$ and $\rho \in (0, 1)$? If yes, how to prove it?

Answer 1

Draw a graph!

The function $u:a\mapsto(L+a)^\rho$ is increasing (this is where one uses that $\rho\gt0$), such that $u(a)\gt a$ for every $a$ in $[0,a^*)$ and $u(a)\lt a$ for every $a$ in $(a^*,+\infty)$ (this is where one uses that $\rho\lt1$), for some $a^*\gt0$ (and even $a^*\gt1$) which is uniquely defined by the identity $$ a^*=u(a^*). $$ Thus, for every starting point $a_0$ in $[0,a^*)$ (such as $a_0=0$), the sequence defined by $a_n=u(a_{n-1})$ for every $n\geqslant1$ is increasing and converges to $a^*$.

Likewise, for every starting point $a_0$ in $(a^*,+\infty)$, the sequence defined by $a_n=u(a_{n-1})$ for every $n\geqslant1$ is decreasing and converges to $a^*$.