Back to Infinite Sequences and Series

Lesson PreviewLesson 5 of 15 in this unit

10.5 Harmonic Series and p-Series

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.

Start free
The harmonic series is
∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}
and it diverges. More generally, a p-series has the form
∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}
A p-series converges if p>1 and diverges if p≤1{p\le1}. This is one of the most important families of benchmark series because many other series are compared to them.

You should memorize the p-series rule because it is used constantly in later convergence tests. The harmonic series is the special case p=1, so it diverges. If the exponent is large enough, meaning p>1, the terms shrink fast enough for the series to converge. If p is too small, the terms do not shrink quickly enough, and the series diverges. This gives you a fast way to classify many standard series immediately.

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-SeriesThis lesson
  6. 10.6Comparison Tests for ConvergencePreview lesson
  7. 10.7Alternating Series Test for ConvergencePreview lesson
  8. 10.8Ratio Test for ConvergencePreview 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

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free