SUMMARY
The series \(\sum_{n=1}^{\infty} \left( \frac{e}{n} \right) ^n\) converges as demonstrated using the Root Test. The limit calculated is \(\lim_{n \to \infty} \sqrt[n]{\left( \frac{e}{n} \right) ^n} = \lim_{n \to \infty} \frac{e}{n} = 0\), which is less than 1, confirming convergence. This method effectively applies the Root Test to establish the behavior of the series.
PREREQUISITES
- Understanding of infinite series and convergence
- Familiarity with the Root Test for series convergence
- Basic knowledge of limits and their properties
- Experience with exponential functions and their behavior
NEXT STEPS
- Study the application of the Ratio Test for series convergence
- Explore other convergence tests such as the Comparison Test
- Learn about power series and their convergence properties
- Investigate the implications of convergence in real analysis
USEFUL FOR
Mathematics students, educators, and anyone studying series convergence in calculus or real analysis will benefit from this discussion.