The formula
P(x) = Σ f⁽ⁿ⁾(a) / n! · (x − a)ⁿ
Why it works
Taylor’s idea is that a polynomial can imitate a function near a point if it matches its value, then its slope, then its curvature and so on: each derivative added pins down one more property and the resemblance stretches a little further. The factorial in the denominator is there to cancel the one that appears when you differentiate a power; without it the coefficients would run away. When the centre is zero the series is called a Maclaurin series, which is nothing more than the commonest case.
How to solve it by hand
- Differentiate the function as many times as the order you want
- Evaluate each derivative at the point you chose as centre
- Divide each value by the factorial of its order
- Multiply by (x − centre) to that power and add it all up
What is worth knowing
A Taylor polynomial only approximates well near the centre, and how far it reaches depends on the function: the one for e^x works on the whole line, while the one for 1/(1−x) breaks as soon as x passes one, however many terms you add. That boundary is the radius of convergence. In practice the third or fourth order approximation is what physics uses to linearise: the simple pendulum works because sin(x) ≈ x for small angles, which is exactly the first term of this series.