An example would be an infinite exponential series of products of geometric series.
$S = \sum_{i=0}^\infty S_n $
$S_n = \exp\{ a \sum_{i=0}^n \rho^i + nC \}= \exp\{a (\frac{1-\rho^{n+1}}{1-\rho}) + nC\}$
Where $\rho\in(0,1)$ and $C<0$
Does a closed-form solution exist? Or is there a reliable approximation method outside of the standard numerical methods (which are impossible for my application)?