Enter a function, a center point, and the order of approximation to expand it as a Taylor (or Maclaurin) series. You get back the polynomial expansion up to your chosen order, plus each term's coefficient and a breakdown table showing derivatives evaluated at the center point. Set the center point to 0 for a Maclaurin series.
Taylor Approximation at x
--
Exact Function Value at x
--
Absolute Error
--
Number of Terms Used
--
A Taylor series is a representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point. It takes the form F(x) = Σ [f⁽ⁿ⁾(a) / n!] · (x − a)ⁿ, where 'a' is the center point and n runs from 0 to infinity. It allows complex functions to be approximated by polynomials.
A Maclaurin series is simply a special case of the Taylor series where the center point a = 0. All Maclaurin series are Taylor series, but not vice versa. If you set the center point to 0 in this calculator, you get the Maclaurin expansion of your chosen function.
The Taylor (Maclaurin) series of e^x centered at a = 0 is: e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + … = Σ xⁿ/n!. This series converges for all real values of x, making it one of the most useful series expansions in mathematics.
First, compute successive derivatives of the function f(x). Second, evaluate each derivative at the center point a. Third, divide each evaluated derivative by the corresponding factorial k!. Finally, multiply each result by (x − a)^k and sum all the terms up to your chosen order n.
The order (n) tells you how many derivative terms to include in the polynomial approximation. A higher order means more terms and a more accurate approximation, but the polynomial becomes more complex. For smooth functions close to the center point, even low-order approximations can be quite accurate.
The absolute error is the difference between the exact function value and the Taylor polynomial approximation at your chosen x value. It shows how close the truncated polynomial is to the true function. The error decreases as you increase the order and as x moves closer to the center point a.
Yes, but the natural logarithm ln(x) requires a center point greater than 0, since ln(0) is undefined. A common choice is a = 1, which gives the expansion ln(x) ≈ (x−1) − (x−1)²/2 + (x−1)³/3 − …. The series converges for x in the interval (0, 2] when centered at a = 1.
The radius of convergence R is the distance from the center point a within which the Taylor series converges to the actual function. For some functions like e^x and sin(x), R is infinite (converges everywhere). For others like 1/(1−x) centered at 0, R = 1, meaning the series only converges for |x| < 1.