Maclaurin Series Calculator
A Maclaurin series approximates a function near zero as a polynomial built from its derivatives at zero. Each term takes a derivative value, multiplies it by a power of x, and divides by the matching factorial. This calculator evaluates the fourth-order Maclaurin polynomial: you supply the function value and its first four derivatives at zero, plus the x at which to evaluate, and it returns the partial sum. Because you provide the exact derivative values, the result is deterministic and never invented. It is ideal for quick approximations near the origin.
Maclaurin series formula
P4(x) = f(0) + f'(0) x + f''(0) x^2 / 2!
+ f'''(0) x^3 / 3! + f''''(0) x^4 / 4!
general term: f^(k)(0) * x^k / k!
2! = 2, 3! = 6, 4! = 24
best accuracy near x = 0
Each term scales a derivative at zero by a power of x and the reciprocal factorial. Summing the first five terms gives the fourth-order Maclaurin approximation of the function.
Notes on Maclaurin series
- The five derivative inputs are the function and its derivatives evaluated at zero.
- Accuracy is highest near x equals zero and falls off farther away.
- The term contributions list shows each term's value, in order.
- A Maclaurin series is a Taylor series centered at zero.
- Many standard functions, such as the exponential, have all derivatives at zero equal to one.
Maclaurin series: frequently asked questions
What is a Maclaurin series?
A Maclaurin series is a Taylor series expanded about zero. It expresses a function as an infinite sum of terms built from the function's derivatives evaluated at zero, each multiplied by a power of x and divided by a factorial.
What is the Maclaurin series formula?
The series is f(0) + f'(0) x + f''(0) x squared over 2 factorial + f'''(0) x cubed over 3 factorial + and so on. The general term is the kth derivative at zero times x to the k, divided by k factorial.
How many terms does this calculator use?
It evaluates the fourth-order Maclaurin polynomial, using the value and the first four derivatives at zero. That gives terms up to x to the fourth power, a common truncation for hand calculations and quick approximations.
Why supply derivatives at zero instead of a function?
The Maclaurin coefficients are exactly the derivatives at zero. Supplying those values keeps the calculation deterministic and exact for any function, without parsing arbitrary symbolic expressions or risking an invented result.
How accurate is a truncated Maclaurin series?
Accuracy is best near x equals zero and degrades as x moves away from zero. Including more terms improves accuracy within the radius of convergence. The remainder after the fourth-order term shrinks as x approaches zero.
Official sources
- NIST Digital Library of Mathematical Functions: Functions of a complex variable (Taylor series).
- NIST Digital Library of Mathematical Functions: Gamma and factorial functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.