The Ali-Cesaro Stolz theorem, introduced in this paper, extends the classical Cesaro-Stolz theorem by addressing cases where the latter does not yield results or provides a "Does Not Existe (DNE) form. This novel theorem leverages the properties of limits and Z-transforms, offering a robust framework for analyzing sequences and series. The theorem is particularly effective in scenarios where traditional approaches fail, providing new insights and tools for mathematical analysis. This paper presents a detailed formulation of the Ali-Cesaro Stolz theorem, including proofs, applications, and a discussion on its implications and advantages over existing methods.