On the Non-Discrete Nature of Infinity: An Algebraic Analysis of Ramanujan's Extended Results and Boundary Contradictions

This paper documents an unconventional journey into the behavior of divergent series. Instead of viewing infinity ($\infty$) as a distant, unreachable limit, we treat it as a tangible, discrete variable within standard algebraic progression formulas. By comparing these common equations with Srinivasa Ramanujan’s well-known regularized results for linear and cubic series, we uncover a striking anomaly: a recurring correction factor of $\Delta \approx -0.0290103234$. This constant suggests a hidden geometric pattern within divergent limits. We ultimately provide a proof by contradiction, showing that treating $\infty$ as a discrete counting number leads to negative irrational roots, which violates the principle that sequence indexes must belong to the Natural Numbers ($\mathbb{N}$). This result was derived by myself during Class 8. I have it both on .tex and .pdf format for easy access.