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Premium experiment • local polynomial approximation

MATHEMATICS • CALCULUS

Taylor & Maclaurin Series Approximation

Add polynomial terms and watch a local Maclaurin approximation converge toward sine, cosine, exponential or arctangent curves.

● TRUE FUNCTION + POLYNOMIALMaclaurin approximation
Approximation
True value
Absolute error
Terms

Reasoning inspector

live model state
Function—
Evaluation x—
Terms—
True value—
Approximation—
Absolute error—
Signed error—
Last term magnitude—
Previous-term error—

Build a function from derivatives

A Maclaurin polynomial uses derivatives evaluated at zero. Increasing the number of terms usually extends the interval over which the polynomial closely follows the original function.

LEARN WITH THE SIMULATION

Understand Taylor & Maclaurin Series Approximation by experimenting.

START HERE

A Taylor series replaces a function near an expansion point with an infinite polynomial whose coefficients encode derivatives there. This laboratory uses Maclaurin expansions, so the expansion point is x = 0, and displays both the true function and a finite truncation. The live error segment and inspector expose approximation value, signed and absolute error, last included term and the error from one fewer term, making convergence something that can be tested rather than assumed.

BUILD THE IDEA

What you should understand first

  • Interpret Maclaurin coefficients as derivative information stored at x = 0.
  • Separate the infinite Taylor series from a finite Taylor polynomial used in computation.
  • Measure how truncation error changes with evaluation point and number of retained terms.
  • Compare functions with infinite radius of convergence to arctangent, whose Maclaurin series has a finite radius of convergence.

TRY IT YOURSELF

Why can adding more terms dramatically improve one function at x = 2 while failing to rescue another?

  1. Choose exp or sin, set x near 2 and increase the number of terms while monitoring absolute error.
  2. Keep a similar x and switch to arctan.
  3. Increase terms again and compare the behavior of the approximation error.
  4. Move arctan back inside |x| < 1 and repeat, then relate the difference to radius of convergence rather than to polynomial degree alone.
Read the visual:

The Maclaurin series for exp, sin and cos converges for all finite x, although more distant points may require more terms. The arctangent Maclaurin series has radius of convergence 1, so outside |x| = 1 adding terms does not make the infinite series converge to arctan.

WHY IT WORKS 01

Derivatives become polynomial coefficients

For a function analytic near zero, the Maclaurin coefficient of xⁿ is f⁽ⁿ⁾(0)/n!. The polynomial therefore packages local derivative information—value, slope, curvature and higher-order behavior—into algebraic terms. Each added term matches another order of local behavior at the expansion point.

WHY IT WORKS 02

Truncation error depends on both degree and distance

Near x = 0, powers of x shrink rapidly and high-order terms often become tiny, so a short polynomial can be excellent. Farther away, the same powers can magnify omitted terms. The last included term is therefore informative but is not by itself a universal error bound for every function and every x.

WHY IT WORKS 03

Arctangent exposes radius of convergence

The Maclaurin expansion arctan(x) = x − x³/3 + x⁵/5 − … has radius of convergence 1 because its nearest complex singularities lie at ±i. Inside |x| < 1 the power series converges; at x = ±1 it converges conditionally; outside that radius its terms do not approach a convergent representation of arctan. This makes it a strong counterexample to the idea that more Taylor terms always help everywhere.

DO NOT MIX THESE UP

Common misconception

A Taylor polynomial is not automatically accurate everywhere, and adding terms is not a universal guarantee of improvement at a point outside the series' interval of convergence. The displayed error belongs to a finite truncation at one selected x; it is not a global error bound. The arctangent case is especially useful because its Maclaurin power series converges only for |x| ≤ 1, with endpoint behavior that differs from interior absolute convergence.

PRACTICE CHALLENGE

Can you meet an error target with the fewest retained terms?

  • Choose a precision target p so the required absolute error is at most 10⁻ᵖ.
  • Pick a function and x, then increase terms until the target is first met.
  • Reduce the term count by one to verify that you found the boundary.
  • Repeat for a point farther from zero, and for arctan both inside and outside its convergence radius.
Reveal the reasoning

The minimum term count depends on the function, target x and requested tolerance. For sin, cos and exp, finite points can be approximated to arbitrary precision with enough terms. For arctan outside |x| = 1, the Maclaurin series itself does not converge to the function, so increasing the truncation degree cannot guarantee success.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • All displayed series are Maclaurin expansions centered at x = 0.
  • Finite sums use standard real-valued series for sin, cos, exp and arctan.
  • The graph clips extreme values for readability while numerical metrics use the un-clipped approximation.

Model limitations

  • Arctangent's Maclaurin series does not converge for |x| > 1 and converges slowly near |x| = 1.
  • Remainder bounds and arbitrary expansion centers are not included in this version.

CONNECT IT

Where this matters

  • Numerical evaluation of transcendental functions
  • Perturbation and local approximation in physics
  • Error reasoning in numerical analysis
  • Power-series solutions and calculus

WORDS THAT MATTER

Key vocabulary

Taylor series
Power series whose coefficients are determined by derivatives of a function at an expansion point.
Maclaurin series
Taylor series expanded specifically about x = 0.
Taylor polynomial
Finite truncation of a Taylor series using a chosen number of terms.
Truncation error
Difference between the true function value and the finite polynomial approximation.
Radius of convergence
Distance from the expansion point within which a power series converges.