SUMMARY
The series \(\sum^{\infty}_{n=1} \arctan(n)\) diverges. The limit \(\lim_{n\rightarrow +\infty} \arctan(n)\) approaches \(\frac{\pi}{2}\), indicating that the series does not converge. The discussion emphasizes the importance of applying the test for divergence as a first step in determining convergence, particularly for series that are not decreasing.
PREREQUISITES
- Understanding of series convergence tests, specifically the test for divergence.
- Familiarity with the arctangent function and its properties.
- Basic knowledge of limits in calculus.
- Concept of increasing and decreasing functions.
NEXT STEPS
- Study the test for divergence in detail and its application to series.
- Explore the properties of the arctangent function, particularly its asymptotic behavior.
- Learn about other convergence tests, such as the integral test and comparison test.
- Investigate proof by induction techniques for establishing properties of functions.
USEFUL FOR
Students and educators in calculus, mathematicians analyzing series convergence, and anyone interested in understanding the behavior of the arctangent function in the context of infinite series.