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10.8 Ratio Test for Convergence

Part of Infinite Sequences and Series in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The ratio test examines the limit of the ratio of successive terms:
L=lim⁡n→∞∣an+1an∣L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|
If L<1, the series converges absolutely. If L>1 or the limit is infinite, the series diverges. If L=1, the test is inconclusive. The ratio test is especially useful for series involving factorials, exponentials, and powers, where successive-term ratios simplify nicely. It is one of the required convergence tests in BC calculus.

You should use the ratio test when terms contain factorial growth, repeated powers, or products that make an+1/an{a_{n+1}/a_n} much simpler than the original term. The absolute value is important because the test is about convergence size, not sign pattern. If the limit comes out less than 1, the terms shrink fast enough for absolute convergence. If it is greater than 1, the terms do not shrink appropriately, so the series diverges.

In this unit

15 lessons

  1. 10.1Defining Convergent and Divergent Infinite SeriesPreview lesson
  2. 10.2Working with Geometric SeriesPreview lesson
  3. 10.3The nth Term Test for DivergencePreview lesson
  4. 10.4Integral Test for ConvergencePreview lesson
  5. 10.5Harmonic Series and p-SeriesPreview lesson
  6. 10.6Comparison Tests for ConvergencePreview lesson
  7. 10.7Alternating Series Test for ConvergencePreview lesson
  8. 10.8Ratio Test for ConvergenceThis lesson
  9. 10.9Determining Absolute or Conditional ConvergencePreview lesson
  10. 10.10Alternating Series Error BoundPreview lesson
  11. 10.11Finding Taylor Polynomial Approximations of FunctionsPreview lesson
  12. 10.12Lagrange Error BoundPreview lesson
  13. 10.13Radius and Interval of Convergence of Power SeriesPreview lesson
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionPreview lesson
  15. 10.15Representing Functions as Power SeriesPreview lesson

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