Calculus : Radius of convergence of a power series

Question

I've the sequence $\left(a_n\right)_{n \in \mathbb{N}}$ define as $a_0=a_1=1$ with $$ \forall n \in \mathbb{N}^{*}, \ a_{n+1}=a_n+\frac{a_{n-1}}{n+1}$$

I need to find the radius of convergence of $\displaystyle \sum_{ n \in \mathbb{N}}a_nx^n$. My trick was to define $$ w_n=\frac{a_{n+1}}{a_n} $$ Hence with the definition $$ w_n=1+\frac{1}{\left(n+1\right)w_{n-1}} $$ I've shown that if $(w_n)_{n \in \mathbb{N}}$ converges then it converges to $1$. How can I show that $(w_n)$ converges ? Any help ?

Answer 1

If $w_n =1+\frac{1}{\left(n+1\right)w_{n-1}} $, then $w_n > 1$.

Therefore $w_n < 1+\dfrac1{n+1}$.

Therefore $w_n \to 1$.

Answer 2

Hint: Show $1\le a_n \le n$ for all $n.$ That will tell you what the ROC is.