What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is centered at any value a. A Maclaurin series is the special Taylor series centered at a=0.

How are Taylor polynomial coefficients calculated?

The coefficient of (x-a)^k is the kth derivative evaluated at a divided by k factorial.

Does a higher order always make the approximation better?

Not everywhere. Accuracy usually improves near the center when the function is sufficiently smooth and its Taylor series converges, but behavior depends on the function, point and convergence interval.

Why can the calculator reject an expansion center?

The function and every derivative needed through the selected order must be real and finite at the center. Logarithms, roots, denominators and nondifferentiable points can violate that requirement.

Does the calculator upload my function?

No. Expression parsing, symbolic differentiation, evaluation, graphing, copying and file creation run locally in the browser.

Taylor and Maclaurin Series Calculator with Steps

Construct a Taylor polynomial about any center or a Maclaurin polynomial about zero. See each derivative, factorial and coefficient, compare the approximation at a point, and graph it against the original function—all locally in your browser.

Enter a function and center

Taylor polynomial

Definition used

\[P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k\]

Calculated polynomial

Choose the function, center and order, then calculate.

A point comparison will appear here when a comparison point is entered.

Derivative and coefficient steps

For each degree k, calculate f⁽ᵏ⁾(a), divide by k!, then multiply by (x−a)ᵏ.

Calculate a polynomial to see every coefficient step.

Function and polynomial graph

Original function Taylor polynomial

Calculate a polynomial to compare it with the original function near the expansion center.

Calculator methodology and limits

Developed and reviewed by: Starlight Robotics calculator team
Last reviewed: 3 August 2026
Method: Parse the expression → propagate truncated Taylor coefficients through each operation → recover derivative values with f⁽ᵏ⁾(a)=k!cₖ → assemble powers of (x−a). Compact symbolic derivative forms are shown when practical.
Numeric display: Derivative values and coefficients are evaluated as browser floating-point numbers and displayed to about 12 significant digits.
Input limits: One real-valued variable, a 300-character expression, finite center and comparison point up to 1,000,000 in magnitude, and order 0–10.
Privacy: Your function, center, derivatives, results, graph and downloaded text remain on your device.

How to use the Taylor and Maclaurin calculator

Enter a function such as exp(x), sin(x), ln(x), or 1/(1+x^2).
Choose Maclaurin to set a=0, or choose Taylor and enter a custom center.
Select an order from 0 through 10. Order n uses derivatives from f(a) through f⁽ⁿ⁾(a).
Optionally enter a comparison point, then calculate. Review the polynomial, coefficient table, actual error and graph.

Taylor polynomial formula

\[P_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n\]

A Maclaurin polynomial uses the same formula with a=0. The calculator constructs the finite polynomial P_n; it does not claim that the corresponding infinite series converges to the function at every point.

Center matters

The polynomial matches the function and its first n derivatives at the center. It is generally most useful near that center.

Order matters

A higher order adds derivative information, but it does not guarantee lower error at every point or outside the interval of convergence.

Domain matters

The function and every required derivative must be real and finite at the center. For example, ln(x) cannot be expanded about 0 over the reals.

Error shown

At the optional point, signed error is f(x)−Pₙ(x). This observed difference is not a symbolic remainder bound.

Worked examples

Maclaurin for eˣ

Every derivative of e^x equals e^x, so every derivative at 0 is 1.

P₄(x)=1+x+x²/2+x³/6+x⁴/24.

Maclaurin for sin(x)

Derivative values at 0 repeat 0, 1, 0, −1.

P₇(x)=x−x³/6+x⁵/120−x⁷/5040.

Taylor for ln(x) at 1

Let h=x−1. The signs alternate after the linear term.

P₄(x)=h−h²/2+h³/3−h⁴/4.

Taylor and Maclaurin calculator FAQs

What is the difference between Taylor and Maclaurin series?

A Taylor expansion uses a general center a. A Maclaurin expansion is precisely the Taylor expansion centered at zero.

How does the calculator find each coefficient?

It propagates a truncated power series through the parsed expression using Taylor-mode automatic differentiation. The degree-k coefficient is mathematically the same as f⁽ᵏ⁾(a)/k!; the table shows that derivative value, factorial and coefficient.

Why is the result called a polynomial rather than the full series?

The calculator stops at the selected finite order. An infinite Taylor series also requires a convergence analysis before it can be identified with the original function away from the center.

Does increasing the order always improve accuracy?

No. It commonly improves local accuracy for analytic functions inside their convergence interval, but the outcome depends on the function, center and evaluation point.

Why is my center rejected?

The function or one of the required derivatives is undefined, non-real or infinite there. Examples include ln(x) at 0, 1/x at 0 and abs(x) at its corner.

Are trigonometric inputs in degrees or radians?

Radians. Standard calculus derivative and Taylor-series formulas for trigonometric functions assume radian measure.

Does my expression leave the browser?

No. All parsing, symbolic differentiation, evaluation, graphing and export happen locally on your device.