The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$.
Another equation is \begin{align} \sum_{N=1}^{\bar N}P(N)x^N=Z, \end{align} where $\bar N\in\mathbb{N}^+$ and $\bar N \gg 1$ and $P(N)$'s also satisfy $0\leq P(N)\leq 1$.
Is there a closed-form solution for any of these two equations (assuming that the Z here is always feasible)?
I presume that you mean to solve for $x$, in terms of $Z$. But if you call $f(x)$ the function described by the series, you’re just looking for $x=f^{-1}(Z)$, the inverse function. If you’re starting with the series for sine, for instance, the “solution” would be the series for $\arcsin(Z)$. It’s certainly true, if $f^{-1}(Z)=\sum_{n=1}^\infty Q_nZ^n$, that there then is a closed-form expression for each $Q_n$ in terms of $P_1,P_2,\cdots,P_n$. It’s a mess, though. Others may be able to give more insight here.