Find rational representation of a power series

Question

I need to find a rational function $\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials, which value is the same as $\sum_{n = 0}^\infty (2n+1)z^{2n}$ on its convergence domain.

I found $\rho=1$, but now I'm stuck. I thought about setting $2n = m$ but I'm pretty sure it would change the series value. I saw in class a way to go from the rational function to the series, but I can't seem to be able to apply it in the opposite direction (it implied factoring the polynomial with its roots $\alpha_1$ and $\alpha_2$, but here we don't know the degree of the polynomials).

I didn't try separating the series in two, but can I even do that?

Thank you guys!

Answer 1

Hint: $$\sum_{n = 0}^\infty (2n+1)z^{2n} = \big(\sum_{n = 0}^\infty z^{2n+1}\big)'$$

$$=\big({z\over 1-z^2}\big)'=...$$

Answer 2

Hints:

• $\sum_{n = 0}^\infty (2n+1)z^{2n}=\biggl(\sum_{n = 0}^\infty z^{2n+1}\biggr)'$,
• $\sum_{n = 0}^\infty z^{2n+1}=z\sum_{n = 0}^\infty (z^2)^{n}$.