Build polynomial approximations and compare exact function values. Inspect coefficients, errors, and convergence across orders. Export tables and graphs for study, review, and sharing.
| Function | x | Terms | Approximation | Exact Value | Absolute Error |
|---|---|---|---|---|---|
| e^x | 1 | 8 | 2.7182539683 | 2.7182818285 | 0.0000278602 |
| sin(x) | 0.5 | 7 | 0.4794270833 | 0.4794255386 | 0.0000015447 |
| ln(1+x) | 0.3 | 8 | 0.2623707429 | 0.2623642645 | 0.0000064784 |
| atan(x) | 0.7 | 9 | 0.6075157667 | 0.6107259644 | 0.0032101977 |
The calculator uses the Maclaurin expansion centered at x = 0. It builds a finite polynomial by summing derivatives evaluated at zero.
General Maclaurin formula: f(x) ≈ Σ from n = 0 to N - 1 of [ f⁽ⁿ⁾(0) / n! ] xⁿ Coefficient form: aₙ = f⁽ⁿ⁾(0) / n! Polynomial approximation: Pₙ(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
This page supports common functions with well-known series. That makes the calculator fast, stable, and useful for checking homework, class notes, symbolic patterns, and numeric approximation accuracy. The coefficient table also shows how each extra term changes the partial sum. When the approximation gets close to the exact value, the absolute and relative errors shrink. When x moves far from zero, more terms are usually needed. For some functions, the interval of convergence matters even more than the number of terms.
Supported forms include e^x, sin(x), cos(x), sinh(x), cosh(x), ln(1+x), 1/(1-x), and atan(x). Each function has a known Maclaurin pattern. The calculator applies that pattern term by term, evaluates the chosen x, and compares the partial sum with the exact function value whenever the exact value is defined. The next omitted term is also displayed as a quick guide to remaining truncation size.
A Maclaurin series is a Taylor series centered at zero. It approximates a function with polynomial terms built from derivatives at x = 0.
Maclaurin polynomials are usually most accurate near zero. As x moves farther away, omitted terms become more important and the error can grow.
The term count tells the calculator how many powers of x to include. More terms often improve accuracy, but improvement depends on convergence.
Some exact expressions have domain limits. For example, ln(1+x) needs x greater than -1, and 1/(1-x) is undefined at x = 1.
It is the first series term left out of the approximation. Its size can help you judge whether the truncation error is likely small.
No. The Maclaurin series for 1/(1-x) converges only when |x| is less than 1. Outside that interval, the polynomial sum does not settle properly.
The table explains each coefficient and partial sum. The graph shows how the polynomial follows the original function over a chosen interval.
Yes. It is useful for checking classwork, exploring convergence, comparing partial sums, and exporting clean records for notes or assignments.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.