Enter a function f(x) and choose the number of terms to get its Maclaurin series expansion — a power series centered at x = 0. The calculator returns the polynomial approximation, lists each term coefficient, and shows a chart comparing the original function to its approximation over a range of x values.
Series Approximation at x
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Exact Value at x
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Absolute Error
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Terms Used
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A Maclaurin series is a special case of the Taylor series where the expansion point is x = 0. It expresses a function as an infinite sum of terms based on the function's derivatives at zero: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … This allows complex functions to be approximated by simple polynomials.
The Taylor series expands a function around any point a, while the Maclaurin series is specifically centered at a = 0. Every Maclaurin series is a Taylor series, but not every Taylor series is a Maclaurin series. When a = 0 is substituted into the Taylor formula, you get the Maclaurin form.
Select your function from the dropdown (e.g. sin(x), e^x, cos(x)), choose the number of terms n, and enter an x value to evaluate the approximation at. The calculator instantly returns the polynomial expansion, the approximation value, the exact value, and the error between them.
The number of terms n determines how many terms of the infinite series are summed. More terms generally give a more accurate approximation, especially for x values farther from 0. For example, sin(x) with 5 terms gives a very accurate result near x = 0 but may diverge for large x.
Any function that is infinitely differentiable at x = 0 can be expanded. Common examples include sin(x), cos(x), e^x, ln(1+x), 1/(1−x), arctan(x), sinh(x), and cosh(x). Functions with singularities at x = 0, like ln(x) or 1/x, cannot be expanded there.
The Maclaurin series converges best near x = 0 — the point of expansion. As x moves further from zero, more terms are needed for an accurate approximation, and some series (like ln(1+x)) only converge within a limited radius. The region where the series converges is called its radius of convergence.
The radius of convergence R is the range of x values for which the Maclaurin series converges to the actual function value. For e^x, sin(x), and cos(x) it is infinite (all real x). For ln(1+x) and 1/(1−x) it is R = 1, meaning the series only reliably works for |x| < 1.
The absolute error is simply the difference |f(x) − P_n(x)|, where f(x) is the exact function value and P_n(x) is the Maclaurin polynomial sum using n terms. A smaller error means the polynomial is a better approximation at that x value with that many terms.