generating functions and the sequence $x_{n+1}=x_{n}+\frac{1}{x_{n}}$

Question

• I start learning about generating functions , so I ask , for example , what all the deduces that a generatingfunctionologist can make for a sequence like :

• $x_{0}= c$ (some constant , say for example 1).

• $x_{n+1}=x_{n}+\frac{1}{x_{n}}$

Answer 1

Not sure that generating functions enter the picture here but the asymptotics of $(x_n)$ is clear: first prove that $x_n\to+\infty$, then note that $x_{n+1}^2=x_n^2+2+1/x_n^2$, and that this implies first that $x_n^2=2n+o(n)$ and then that $x_n^2=2n+\frac12\log n+o(\log n)$.