Plausible and Genuine Extensions of L'Hospital's Rule

摘要

Let f and g be differentiable real valued functions. Motivated by L'Hospital's Rule, we might expect the convergence of∑f'(n)/g'(n) to imply the convergence of∑f'(n)/g(n) when f, g, f', g' all have limit 0 as x tends to infinity and also when all four functions have infinite limits at infinity. We find this to be true, subject to some very mild additional conditions, when the four functions have limit zero, but not necessarily to be true in the infinity case. For limits, the discrete analogue of L'Hospital's Rule is the Stolz-Cesaro Theorem. We also find a result for series that is in the spirit of the Stolz-Cesaro Theorem.