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.
Tips
- Write out the first few terms to check the pattern before using a formula.
- Decide whether the sequence starts at term 0 or term 1 – formulas differ by one place.
- Arithmetic sequences add a constant; geometric sequences multiply by one.
- An infinite geometric series only has a sum when |r| < 1.
Common mistakes to avoid
- Off-by-one errors in the term number.
- Using the arithmetic sum formula for a geometric sequence.
- Ignoring the sign of a negative ratio or difference.
Inputs explained
| Field | What to enter |
|---|---|
| Degree of the Taylor polynomial n | A number |
| Point x | A number |
| Centre a | A number |
| Bound 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 n | Error bound |
|---|---|
| 2.5 | 0.1667 |
| 3.75 | 0.0417 |
| 5 | 0.0014 |
| 6.25 | 0.0002 |
| 7.5 | 2.4802e-5 |
| 10 | 2.5052e-8 |
Formula
|Rₙ(x)| ≤ M·|x − a|ⁿ⁺¹ / (n + 1)!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.
Frequently asked questions
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.
What formula does the Lagrange Error Bound Calculator use?
|Rₙ(x)| ≤ M·|x − a|ⁿ⁺¹ / (n + 1)!.