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Taylor Series Calculator

Calculate Taylor and Maclaurin polynomials for common functions. Choose an expansion center and polynomial degree, inspect the coefficients, and optionally compare the approximation with the actual function value.

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Calculate a Taylor or Maclaurin polynomial

Select a supported function, enter its expansion center and choose the degree. Enter an evaluation point if you also want to estimate the function value and calculate the approximation error.

Use a = 0 for a Maclaurin polynomial.
Leave blank if you only need the polynomial and coefficient table.
Reset

What is a Taylor series?

A Taylor polynomial approximates a function near a chosen center using its derivatives at that point. A Maclaurin polynomial is the special case where the center is zero.

Tn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + ... + f(n)(a)(x-a)n/n!

Supported functions and example expansions

Exponential function e^x = 1 + x + x²/2! + x³/3! + ... The expansion about zero converges for every real x.
Sine function sin(x) = x - x³/3! + x⁵/5! - ... The derivatives repeat in a four-step cycle.
Cosine function cos(x) = 1 - x²/2! + x⁴/4! - ... Odd-power coefficients vanish at the Maclaurin center.
Natural logarithm ln(1+x) = x - x²/2 + x³/3 - ... The standard expansion is centered at zero and is valid locally for |x| less than 1.
Geometric function 1/(1-x) = 1 + x + x² + x³ + ... This geometric power series converges when |x| is less than 1.
Square root sqrt(1+x) = 1 + x/2 - x²/8 + ... Its coefficients follow the generalized binomial expansion.

How to use the calculator

  1. Select the function you want to approximate.
  2. Enter the center of expansion. Choose zero for a Maclaurin polynomial.
  3. Choose a polynomial degree between 0 and 12.
  4. Optionally enter a target x-value to compare the polynomial with the actual function.
  5. Click Calculate Taylor Series to see the polynomial and its coefficients.

Understanding approximation error

The absolute error is |f(x) - Tn(x)|. Increasing the polynomial degree often improves an approximation near its center, but it does not guarantee improvement at every point. Convergence depends on the function and the distance from the expansion center. The error reported here is calculated from the actual function value and the finite Taylor polynomial; it is not an estimate of the error over an entire interval.

Frequently asked questions

What is the difference between Taylor and Maclaurin series?

A Maclaurin series is a Taylor series centered at zero. A Taylor series can be centered at any point where the required derivatives exist.

Can I enter any function?

This version supports eight predefined functions: e^x, sin(x), cos(x), ln(x), ln(1+x), 1/(1-x), sqrt(x), and 1/x. It does not parse arbitrary symbolic expressions.

Why is my center rejected?

Some functions or derivatives are undefined at particular points. For example, ln(x) requires a positive center, sqrt(x) requires a positive center for a Taylor expansion, and 1/x cannot be expanded at zero.

Does a higher degree always give a better approximation?

Not necessarily. Taylor polynomials are generally local approximations. Their accuracy depends on the function, the center, the evaluation point and the convergence properties.

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