How to use this calculator
Type a function, the center a, and the order n (1 to 10). The calculator differentiates n times, evaluates each derivative at the center, divides by k! to get the coefficients, and assembles the polynomial. Enter a point to see how close the polynomial comes to the real function there.
Formula
| Function | Maclaurin series (a = 0) |
|---|---|
| ex | 1 + x + x²/2! + x³/3! + … |
| sin x | x − x³/3! + x⁵/5! − … |
| cos x | 1 − x²/2! + x⁴/4! − … |
| 1/(1 − x) | 1 + x + x² + x³ + … (|x| < 1) |
| ln(1 + x) | x − x²/2 + x³/3 − … (−1 < x ≤ 1) |
Worked example
eˣ to 4th order at a = 0, evaluated at x = 0.5 (the default)
- Every derivative of eˣ is eˣ, so f(k)(0) = 1 and ck = 1/k!.
- P₄(x) = 1 + x + x²/2 + x³/6 + x⁴/24.
- P₄(0.5) = 1 + 0.5 + 0.125 + 0.0208333 + 0.0026042 = 1.6484375, against e0.5 = 1.6487213, an error of 0.00028.
Frequently asked questions
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is a Taylor series centered at a = 0. Everything else is the same.
Why does the polynomial fail far from the center?
A polynomial matches the function’s derivatives only at the center. Far away the remainder grows, and some series (like 1/(1 − x)) only converge inside a radius of convergence.
Where are Taylor series used in engineering?
Small-angle approximations (sin θ ≈ θ), linearizing nonlinear systems around an operating point, error estimates in numerical methods, and deriving finite-difference formulas all come from Taylor series.