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Intermediate Topic 34 of 51 Math for Computing

Taylor Series

Approximating functions by polynomials.

How it works

A Taylor series approximates a smooth function near a point as an infinite polynomial built from its derivatives: f(x) equals f(a) plus f'(a)(x minus a) plus f''(a)(x minus a) squared over 2 factorial and so on. Truncating after n terms gives a Taylor polynomial with a remainder bounding the error. The Maclaurin series is the case at zero, with well-known expansions for e to the x, sine, cosine and 1 over (1 minus x). The radius of convergence tells where the series represents the function.

Where it is used

Computing functions in calculators and math libraries, linearizing nonlinear systems for Newton's method and optimization, analyzing algorithms and approximations in physics and engineering use Taylor series.

Common pitfall

The approximation is good only near the expansion point, and the series may converge only within a limited radius. Truncation error must be estimated. Alternating sign cancellation causes numerical precision loss. Not every infinitely differentiable function equals its Taylor series. Use range reduction and more efficient approximations, such as minimax polynomials, for production code.

Where it fits in the course

Taylor Series is topic 34 of 51 in Math for Computing, rated Intermediate. It follows Integrals. The next topic is Optimization.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Math for Computing quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Taylor Series?
Approximating functions by polynomials. It belongs to Math for Computing and is an intermediate-level topic.
Where do people use Taylor Series?
Computing functions in calculators and math libraries, linearizing nonlinear systems for Newton's method and optimization, analyzing algorithms and approximations in physics and engineering use Taylor series.
What is the most common mistake with Taylor Series?
The approximation is good only near the expansion point, and the series may converge only within a limited radius.
What should I learn before this topic?
Start with Integrals, the topic just before it in Math for Computing.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Math for Computing quiz.
Is Taylor Series hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Taylor Series?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Taylor Series?
Integrals, Optimization, Partial Derivatives and Gradients are closely connected. The full list is in the related topics section of this page.
What should I study after Taylor Series?
Continue with Optimization, the next topic in Math for Computing.

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