SUMMARY
The series defined by the terms n!/e^n diverges. The ratio test was applied, specifically using the limit lim n-->infinity [(n+1)!/(e^(n+1))][(e^n)(n!)]. This simplification led to the conclusion that as n approaches infinity, the limit approaches infinity, confirming that the series does not converge. Therefore, the final determination is that the series diverges.
PREREQUISITES
- Understanding of the ratio test in calculus
- Familiarity with factorial notation (n!)
- Knowledge of limits and their properties
- Basic concepts of series convergence and divergence
NEXT STEPS
- Study the application of the ratio test in different series
- Learn about other convergence tests, such as the root test and comparison test
- Explore the behavior of factorial functions in limits
- Investigate the implications of series divergence in mathematical analysis
USEFUL FOR
Students studying calculus, particularly those focusing on series and sequences, as well as educators teaching convergence tests in mathematical analysis.