Hello!

I started to study a differential equation, and turned the problem into a difference equation ($ \displaystyle a_0, a_1 \in \mathbb{R}, a_2 = 0, a_3 = \frac{a_0 + a_1}{6}$)

$ \displaystyle a_{v+2} = \frac{v^2a_v + a_{v-1}}{(v+1)(v+2)}$, where $ \displaystyle v \ge 2$.

The numbers $ \displaystyle a_v$ are coefficients of a serie solution to the original differential equation.

But now the question: I'd like to prove if the serie converges or not (i.e. is it usefull at all). But how is this done in this case? As far as I can see, the difference equation has not err... a nice solution and so all I have is the difference equation. Sometimes I've seen, someone takes a limit of an equation and assumes $ \displaystyle a_{\infty} = a$ and then substituting this into the equation. But this method clearly fails in this case. Are there another methods?

Thank you! ^^