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-...
Advanced Topics

Convergence and Error in Taylor Series

How Accurate Is a Taylor Series?

Taylor series are awesome, but they only approximate functions well near the center point. The further away you go, the bigger the error.

Radius and Interval of Convergence

  • Radius of Convergence: The range where the series converges to the function.
  • Interval of Convergence: The actual \(x\)-values for which the series works.

Error Estimation

  • The remainder or error tells you how close your approximation is.
  • Lagrange Error Bound: Estimates the error from using a partial sum.

\[ |R_n(x)| \leq \frac{M|x-c|^{n+1}}{(n+1)!} \] where \(M\) is the maximum value of the next derivative.

Why Is This Useful?

Engineers and scientists use error bounds to know how many terms are needed for accurate calculations.

Examples

  • Using three terms of the cosine Maclaurin series to estimate \(\cos(0.1)\).

  • Finding the interval where the series for \(\ln(1+x)\) converges.

In a Nutshell

Taylor series are only as good as their convergence and error bounds allow—knowing these keeps math (and science!) reliable.

Next
Modeling Motion with Parametric Equations