AP® Calculus BC · Unit 10 · Topic 10.14
Finding Taylor or Maclaurin Series for a Function
A Taylor series rebuilds a function from its derivatives at one point, using the coefficient formula fn(a)/n! for each power of (x-a). It equals the original function only where the remainder term shrinks to zero.
A Taylor series is built so that it matches a function perfectly at one point: same value, same slope, same curvature, same every derivative, all at once. The coefficient of (x-a)n is fn(a)/n!, the nth derivative at the center divided by n factorial. That factorial is what keeps each term from exploding as n grows, and it's exactly the scaling needed so that differentiating the series n times and plugging in x = a gives back fn(a). Stack infinitely many of these matching conditions and, where it converges to f, the polynomial stops being an approximation and becomes the function.
For many functions you don't need to grind through derivatives. 1/(1+x2) is a geometric series in disguise: treat it as 1/(1-(-x2)) with ratio r = -x2, giving sum of (-1)n x2n, valid for |x| < 1. For ex centered at x = 2, every derivative of ex is ex itself, so fn(2) = e2 for all n, and the series is sum of e2 (x-2)n / n!. Recognizing geometric, known Maclaurin series, or patterns in repeated differentiation saves enormous time.
Having a formula for a Taylor series doesn't guarantee it equals f(x) away from the center. That's where the Lagrange error bound comes in: it bounds the remainder after n terms, and if that bound goes to zero as n grows, the series genuinely converges back to f on that interval. This connects directly to radius and interval of convergence from earlier in the unit, and sets up power series representations and error estimation as the series gets used to approximate values and integrals.
Key terms
- Taylor series
- An infinite polynomial centered at a that matches a function and all its derivatives at that point.
- Maclaurin series
- A Taylor series centered at x = 0.
- Lagrange error bound
- A bound on how far a Taylor polynomial's value can be from the true function value after n terms.
- Radius of convergence
- The distance from the center within which the series is guaranteed to converge.
- nth derivative coefficient
- The coefficient fn(a)/n! that determines the (x-a)n term of a Taylor series.
Watch it explained
- How it worksHow does the general term built from the nth derivative over n factorial generate every coefficient?
- Worked exampleFind the Maclaurin series for f(x) = 1/(1 + x2) in general form.
- Worked exampleFind the Taylor series for f(x) = ex centered at x = 2.
- Common mistakeWhy doesn't every infinitely differentiable function equal its own Taylor series?
- Ties togetherHow does the Lagrange error bound decide whether this Taylor series actually converges back to the function that generated it?
Check yourself
End of topic.