AP Calculus BC

Advanced Placement Calculus BC including series, parametric equations, and polar functions.

Sequences and Series IntroductionConvergence TestsPower Series and Taylor Series
Parametric Equations and CalculusPolar Coordinates and CalculusConvergence and Error in Taylor Series
Modeling Motion with Parametric EquationsSignal Processing with SeriesEngineering with Polar Functions
Mastering Series QuestionsVisualizing Parametric and Polar CurvesBreaking Down Multi-Step Problems
Sequences and Series...Convergence TestsPower Series and Tay...Parametric Equations...Polar Coordinates an...Convergence and Erro...Modeling Motion with...Signal Processing wi...Engineering with Pol...Mastering Series Que...Visualizing Parametr...Breaking Down Multi-...
Basic Concepts

Convergence Tests

How Do You Know If a Series Converges?

Not all infinite series have a sum! Calculus gives us special tests to decide if a series converges (adds up to a finite number) or diverges (gets infinitely large).

Popular Convergence Tests

  • The nth-Term Test: If the terms don't approach zero, the series diverges.
  • Geometric Series Test: If the ratio \(|r| < 1\), the series converges.
  • Integral Test: Compare a series to an improper integral.
  • Comparison Test: Compare your series to a known converging/diverging series.
  • Ratio and Root Tests: Great for series with factorials or powers.

Why Do We Care?

Knowing if a series converges is crucial in physics, engineering, and even finance, where infinite processes are modeled.

Summary Table

TestWhen to Use
GeometricSeries of the form \(ar^n\)
Ratio/RootFactorials and powers
ComparisonSeries with similar known series

Examples

  • The series \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) passes the geometric test and converges.

  • The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) (the harmonic series) diverges by the nth-term test.

In a Nutshell

Convergence tests help us decide if an infinite series adds up to a real number.

Next
Power Series and Taylor Series