AP Calculus BC

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

Limits and ContinuityDifferentiation and ApplicationsIntegration and Its Uses
Infinite Series and ConvergenceParametric Equations and MotionPolar Coordinates and Graphs
Analyzing Population GrowthDesigning Roller Coasters with Parametric EquationsSignal Processing and Polar Graphs
Mastering Multiple-Choice QuestionsEffective Free-Response Techniques
Limits and Continuit...Differentiation and ...Integration and Its ...Infinite Series and ...Parametric Equations...Polar Coordinates an...Analyzing Population...Designing Roller Coa...Signal Processing an...Mastering Multiple-C...Effective Free-Respo...
Advanced Topics

Infinite Series and Convergence

Summing the Infinite

An infinite series adds up terms forever! But not all series add to a finite number. Understanding convergence means knowing when an infinite sum makes sense.

  • A series converges if the sum approaches a real number as more terms are added.
  • The geometric series \( \sum_{n=0}^\infty ar^n \) converges if \( |r| < 1 \).
  • Tests like the Ratio Test, Root Test, and Integral Test help determine convergence.

Real-World Connections

Infinite series are key for representing complicated functions, like \( e^x \) or \( \sin x \), as power series.

Key Formula

\[S = \frac{a}{1 - r}\]

Examples

  • The sum \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots = 2 \).

  • The series \( \sum_{n=1}^\infty \frac{1}{n^2} \) converges to \( \frac{\pi^2}{6} \).

In a Nutshell

Infinite series can sum up forever—if they're convergent, the total is finite!

Key Terms

Convergence
When the sum of all terms in a series approaches a specific value.
Power Series
An infinite series written as \( \sum_{n=0}^\infty a_n (x-c)^n \).
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Parametric Equations and Motion
Infinite Series and Convergence - AP Calculus BC Content | Practice Hub