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Calculus · Series

Taylor Polynomial & Approximation Error Calculator

Build Maclaurin approximations for exponential, sine, cosine, and ln(1+x), then compare the polynomial value with the reference function.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Maclaurin approximation centered at x = 0

Polynomial

Tₙ(x)=Σ f⁽ᵏ⁾(0)xᵏ/k!

Approximation

f(x)≈Tₙ(x)

Observed error

E=|f(x)−Tₙ(x)|
Calculus · Approximation

Build a Maclaurin polynomial and measure error

This reviewed version uses Maclaurin polynomials centered at 0. For ln(1+x), x must be greater than −1.

Inputs and displayed decimal results use at most three decimal places. Full internal precision is retained during calculation.

Tₙ(x)=Σ f⁽ᵏ⁾(0)xᵏ/k!

Choose a supported function, evaluation point, and degree.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The polynomial is centered at zero and limited to degree 12. The displayed error is the observed absolute difference at the entered x, not a general symbolic remainder bound. ln(1+x) requires x>−1.

Worked examples

See the method in practice

01

Approximate e

The degree-8 Maclaurin polynomial for eˣ at x=1 is already close to e.

02

Approximate sine

Near zero, x−x³/6+x⁵/120 provides a useful approximation to sin x.

Questions

Frequently asked

What is a Maclaurin polynomial?

It is a Taylor polynomial whose center is x=0.

Does a higher degree always help?

Usually near the center, but accuracy also depends on the function, evaluation point, and convergence region.

Is the reported error a proof bound?

No. It is the numerical absolute difference at the chosen point.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Calculus Volume 2 — Taylor and Maclaurin Series

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology