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Taylor and Maclaurin Series
See how Taylor and Maclaurin series rewrite functions as infinite power series, with formulas, worked examples for sin x and ln x, and convergence tips.
What You'll Learn
Write the Taylor series formula using derivatives of f evaluated at a point a
Recognize the Maclaurin series as the special case of a Taylor series centered at a equals zero
Recall the standard series expansions for e to the x, sin x, cos x, and 1 over 1 minus x
Compute a Maclaurin series by finding successive derivatives at zero and dividing by factorials
Build a Taylor series centered at a nonzero point such as ln x expanded about a equals 1
Check the interval of convergence of a power series before using it to approximate a function
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
PEI Curriculum Aligned