# Taylor Series Calculator - Taylor & Maclaurin Polynomials with Error and Graph

> Free online Taylor series calculator: exact coefficients for f(x) at centre a up to order 20, as fractions, with truncation error and a plot against f.

- URL: https://qilingtools.com/en/math/taylor
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

A Taylor expansion writes a function near a point as a polynomial: f(x) ≈ f(a) + f′(a)(x−a) + f″(a)/2!·(x−a)² + …. Enter the function and the centre and you get the Taylor polynomial of order n, every coefficient (as a fraction such as 1/6 or −1/5040 where possible), the true error at a point of your choice, and a plot of the function together with several orders so you can watch the approximation improve. Centre 0 gives the Maclaurin series.

## How to use

1. Enter f(x), for example sin(x), e^x, ln(1+x) or 1/(1-x)
2. Enter the centre a (expressions like pi/4 work) and drag to choose the order n
3. Read the Taylor polynomial, the coefficient table and the radius estimate
4. Enter a check point x to see the error, pick orders to compare on the plot, or open it in the grapher

## Features

- Forward-mode automatic differentiation on truncated power series — no difference quotients, so even order 20 stays exact
- Handles sin, cos, tan, asin, atan, sinh, exp, ln, lg, sqrt, cbrt, powers, x^x and any composition of them
- 0/0 forms such as sin(x)/x or (e^x−1)/x expand directly at 0
- Coefficients shown as fractions or factorial forms such as −1/5040, 1/19! or −5/128
- Exact f(x), Pₙ(x), absolute and relative error, and the size of the next term
- Radius of convergence estimated from the coefficients, with a warning when the check point lies outside it
- Inline plot of the function against several Taylor polynomials, plus a link to the full grapher
- Copy the polynomial as an expression the grapher understands

## FAQ

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

A Maclaurin series is simply a Taylor series centred at a = 0, e.g. e^x = 1 + x + x²/2! + x³/3! + … and sin x = x − x³/3! + x⁵/5! − …. Enter 0 as the centre to get it.

### How are the coefficients computed, and are they exact?

x is written as a + t and every operation in the expression — arithmetic, powers, trig, exponentials, logarithms — is carried out on power series in t. This is forward-mode automatic differentiation: the derivatives come from exact recurrences with only floating-point rounding, unlike finite differences, which fall apart at high orders.

### Why does a higher order make the error worse far from the centre?

A Taylor series only converges inside its radius of convergence. For 1/(1−x) or ln(1+x) around 0 the radius is 1, so at x = 1.5 adding terms makes things worse, not better. The tool estimates the radius and warns you when your check point is outside it.

### Which functions cannot be expanded?

Functions that are undefined or not differentiable at the centre, such as ln x and √x at 0, |x| at 0 or tan x at π/2. Pick another centre instead — ln x expands nicely around a = 1.

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