Taylor Series
Represent functions with infinite polynomials.
Key idea
Represent functions with infinite polynomials.
In depth
f(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!. It builds a function from its derivatives at one point. Truncating gives polynomial approximations that improve as more terms are added.
Worked example
eˣ = 1 + x + x²/2! + x³/3! + ... At x = 1: e ≈ 1 + 1 + 0.5 + 0.1667 + 0.0417 ≈ 2.708.
Formula
Real-world uses
- Approximating sin, exp and other functions in calculators, computer code and physics
Common mistake
Skipping the factorial in the denominator.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Taylor Series?
Represent functions with infinite polynomials. It is a topic at the expert level in Calculus.
What is the formula for Taylor Series?
The key formula is f(x) = Σ f⁽ⁿ⁾(a)/n! (x−a)ⁿ.
Where is Taylor Series used in real life?
Real-world examples: Approximating sin, exp and other functions in calculators, computer code and physics.
What should I learn before Taylor Series?
Make sure you are comfortable with Power Series, the previous topic in Calculus.
What is a common mistake with Taylor Series?
Skipping the factorial in the denominator.
How can I practise Taylor Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Taylor Series?
Move on to Common Maclaurin Series next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.