Series Convergence Tests to Know for AP Calculus AB/BC

Understanding series convergence tests is essential in calculus. These tests help determine whether a series converges or diverges, guiding you through various methods like the Divergence Test, Integral Test, and more, each with unique applications and criteria.

1. Divergence Test

   • If the limit of the series' terms does not approach zero, the series diverges.
   • This test is a necessary condition for convergence; if it fails, the series cannot converge.
   • It is often the first test to apply when analyzing a series.

2. Integral Test

   • Applicable to positive, continuous, and decreasing functions.
   • If the integral of the function converges, then the series converges; if the integral diverges, so does the series.
   • Useful for series that can be expressed as the sum of function values.

3. Comparison Test

   • Compares a given series to a known benchmark series.
   • If the series being tested is less than a convergent series, it converges; if it is greater than a divergent series, it diverges.
   • Requires careful selection of the comparison series.

4. Limit Comparison Test

   • Compares the limit of the ratio of two series' terms.
   • If the limit is positive and finite, both series either converge or diverge together.
   • Useful when direct comparison is difficult.

5. Ratio Test

   • Analyzes the limit of the absolute value of the ratio of consecutive terms.
   • If the limit is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
   • Particularly effective for series involving factorials or exponentials.

6. Root Test

   • Examines the limit of the nth root of the absolute value of the terms.
   • Similar to the Ratio Test, it provides convergence criteria based on the limit being less than, greater than, or equal to 1.
   • Useful for series with terms raised to the nth power.

7. Alternating Series Test

   • Specifically for series whose terms alternate in sign.
   • If the absolute value of the terms decreases to zero, the series converges.
   • Provides a straightforward method for determining convergence of alternating series.

8. Absolute Convergence Test

   • If the series of absolute values converges, then the original series converges absolutely.
   • Absolute convergence implies convergence, which is a stronger condition.
   • Important for determining the behavior of series with alternating terms.

9. p-Series Test

   • A specific case of the Comparison Test for series of the form 1/n^p.
   • Converges if p > 1 and diverges if p ≤ 1.
   • Provides a quick reference for many common series.

10. Telescoping Series

    • A series where most terms cancel out when expanded.
    • Often results in a finite sum, making it easier to evaluate convergence.
    • Useful for series that can be expressed in a form that reveals cancellation.