This research presents a comprehensive investigation into the convergence analysis of approximate solutions for the generalized one-dimensional linear telegraph equation, utilizing the Reproducing Kernel Hilbert Space (RKHS) method. The primary objective of this study is to establish a robust mathematical framework that bridges the gap between numerical approximations and exact analytical solutions. By employing a sophisticated iterative procedure within the RKHS environment, we rigorously demonstrate that the generated approximate results exhibit a strong and consistent convergence behavior toward the exact solutions derived from Fourier series expansions. The theoretical foundations of the RKHS technique allow for a systematic treatment of the telegraph equation, which is a fundamental model in various physical and engineering phenomena, such as signal propagation and vibration analysis. To evaluate the practical performance and reliability of the proposed methodology, several numerical experiments were conducted. The empirical results consistently highlight the method's exceptional precision, stability, and computational integrity. Furthermore, the study underscores a remarkably rapid convergence rate, signifying that high-fidelity results can be achieved with minimal computational effort compared to traditional numerical schemes. These findings validate the RKHS method as a highly efficient and potent alternative for solving linear partial differential equations, confirming its significant potential for broader applications in the field of applied mathematics and numerical analysis.