AP Calculus BC
Taylor Polynomial vs Taylor Series
A Taylor polynomial is a finite truncation, so it approximates the function and carries an error you can bound. A Taylor series continues forever and, inside its interval of convergence, equals the function exactly.
Taylor polynomial
Use when: The question names a degree, asks for an approximation, or asks you to bound an error.
Taylor series
Use when: The question asks for the full series, a general term, or an interval of convergence.
Side by side
| Polynomial | Series | |
|---|---|---|
| Number of terms | Finite | Infinite |
| Relation to | Approximates | Equals, inside the interval |
| Has an error | Yes | No, inside the interval |
| Error bound | Lagrange, or first omitted term | Not applicable |
The degree counts the highest power actually present, which can be lower than the number of terms you generated. For only odd powers appear, so the second and third degree polynomials are the same expression.
Error questions always concern the polynomial, never the series. Use the Lagrange error bound in general, or the first omitted term when the series alternates, which is much quicker.
The coefficient trap
The coefficient of is the th derivative at the centre divided by . Dropping the factorial is the most common slip in building either object.
Frequently asked questions
Does a Taylor series always equal the function?
Inside its interval of convergence, for the functions in AP Calculus, yes. Outside the interval the series diverges and says nothing.
In the CED: Unit 10: Infinite Sequences and Series (BC)