# Lagrange Error Bound Calculator

URL: https://speedycalculator.com/calculators/math/lagrange-error-bound/
Last updated: 2026-10-30

> The Lagrange error bound calculator finds the maximum error of an n-th degree Taylor polynomial centred at a, at a point x, given a bound M on the (n + 1)-th derivative.

## Inputs

- Degree of the Taylor polynomial n
- Point x
- Centre a
- Bound M on |f⁽ⁿ⁺¹⁾|

## Results

- Error bound
- (n + 1)!

## Formula

```
|Rₙ(x)| ≤ M·|x − a|ⁿ⁺¹ / (n + 1)!
```

## Explanation

Taylor polynomials approximate functions such as sin x or eˣ near a point. The Lagrange error bound tells you how many terms you need for a given accuracy – for example to compute e or π to a set number of digits.

### How to use the Lagrange Error Bound Calculator

- **Degree of the Taylor polynomial n**
- **Point x**
- **Centre a**
- **Bound M on |f⁽ⁿ⁺¹⁾|**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the degree n, the centre a, the point x and M, the largest value of |f⁽ⁿ⁺¹⁾| between a and x. For sin and cos, M = 1 works everywhere.

### Inputs explained

FieldWhat to enterDegree of the Taylor polynomial nA numberPoint xA numberCentre aA numberBound M on |f⁽ⁿ⁺¹⁾|A number

### What each result means

- **Error bound** = [Bound M on |f⁽ⁿ⁺¹⁾|]×abs([Point x] - [Centre a])^([Degree of the Taylor polynomial n] + 1)÷fact([Degree of the Taylor polynomial n] + 1)
- **(n + 1)!** = fact([Degree of the Taylor polynomial n] + 1)

### Example values

How the main result changes when **Degree of the Taylor polynomial n** changes and everything else stays at the example values above:

Degree of the Taylor polynomial nError bound2.50.16673.750.041750.00146.250.00027.52.4802e-5102.5052e-8

## Worked example

Inputs — Degree of the Taylor polynomial n: 5; Point x: 1; Centre a: 0; Bound M on |f⁽ⁿ⁺¹⁾|: 1.

Results — **Error bound = 0.001388888889**; (n + 1)! = 720.

## Key terms

- **Remainder**: The difference between a function and its Taylor polynomial.

## FAQ

**What is M?**
The maximum of the absolute (n + 1)-th derivative on the interval between a and x.

**How many terms do I need for e to 6 decimals?**
n = 9 gives error < 3/10! ≈ 8 × 10⁻⁷.

**Is the bound the actual error?**
No – it is an upper limit; the true error is usually smaller.

**What is a Maclaurin polynomial?**
A Taylor polynomial centred at a = 0.


## Sources

- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)
- [Wolfram MathWorld – Lagrange remainder](https://mathworld.wolfram.com/LagrangeRemainder.html)