SUMMARY
The convergence of the series ∑ tan(1/k) for k=5 can be established using the comparison test. The discussion highlights that the ratio test yields an inconclusive result of 1, and the divergence test results in 0, indicating that neither method is effective alone. By applying the inequality sin(x) ≥ (2x/π) for 0 < x ≤ π/2, it is shown that tan(1/k) is greater than or equal to (2/πk), which leads to the conclusion that the series converges.
PREREQUISITES
- Understanding of series convergence tests, specifically the comparison test.
- Familiarity with trigonometric functions, particularly the behavior of tan(x) near zero.
- Knowledge of limits and inequalities in calculus.
- Experience with the ratio test and divergence test for series.
NEXT STEPS
- Study the comparison test in detail, focusing on its application in series convergence.
- Explore the properties of trigonometric functions and their limits as arguments approach zero.
- Learn about the implications of the ratio test and when it is inconclusive.
- Investigate other convergence tests such as the integral test or the root test.
USEFUL FOR
Students studying calculus, particularly those focusing on series convergence, as well as educators seeking to explain convergence tests and trigonometric series behavior.