HallsofIvy said:It should be clear that for n> 3, n^2> n!.
The purpose of this proof is to determine the convergence of a factorial series without relying on the Ratio Test, which may be difficult or impossible to use in certain cases.
Convergence of factorial can be proven using other methods such as the Comparison Test, the Root Test, or the Integral Test.
The Ratio Test may not work for all factorial series, especially those with alternating signs or with terms that do not tend to zero. It also requires the limit of the ratio of consecutive terms to be taken, which can be time-consuming or challenging to calculate in some cases.
By using alternative methods to prove convergence, we can gain a better understanding of the behavior of the factorial series and potentially find simpler or more efficient ways to prove convergence.
Yes, the Comparison Test, Root Test, and Integral Test can be applied to other types of series as well to determine their convergence. However, the specific techniques used may vary depending on the type of series being analyzed.