Chapter 8.15
Taylor series and Maclaurin series
What You'll Learn
Express functions as Taylor series centered at any point or Maclaurin series at zero
Calculate the nth derivative and recognize patterns in successive derivatives
Apply Taylor and Maclaurin series formulas to exponential, sine, and cosine functions
Manipulate known series formulas by substitution to find new series representations
Verify factored results and check series convergence by expanding terms
What You'll Practice
1
Finding Maclaurin series for e^(2x) by computing nth derivatives and recognizing patterns
2
Determining Taylor series for polynomials by expanding terms until derivatives become zero
3
Using substitution to find series for composite functions like e^(x²) and sin(4x)
4
Deriving Taylor series for sine and cosine from scratch using derivative patterns
Why This Matters
Taylor and Maclaurin series are fundamental tools in calculus that let you approximate complex functions with polynomials, making them essential for solving differential equations, modeling physics problems, and performing numerical computations in engineering and computer science.
Before You Start — Make Sure You Can:
This Unit Includes
7 Video lessons
Practice exercises
Learning resources
Skills
Taylor Series
Maclaurin Series
Power Series
nth Derivative
Series Expansion
Pattern Recognition
Trigonometric Series
