Last updated: July 31, 2026
Lagrange Error Bound Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
R_n(x) ≤ (M/(n+1)!) × |x-a|^(n+1)
Where:
- M=Maximum value of |f^(n+1)(c)| on the interval
- n=Taylor polynomial degree
- x=Evaluation point
- a=Center of expansion
- R_n(x)=Error bound
Worked Examples
Fourth-derivative bound example
Use M = 2, n = 3, x = 1.5, and a = 1.
- 1Compute (n+1)! = 4! = 24.
- 2Compute |x-a| = |1.5-1| = 0.5.
- 3Raise to the fourth power: 0.5⁴ = 0.0625.
- 4Bound = 2 × 0.0625 / 24 ≈ 0.005208.
Exact center gives zero bound contribution
If x equals a, the power term is zero.
- 1Compute |x-a| = 0.
- 2Any positive power of 0 is 0.
- 3The bound becomes 0.
Invalid degree
The degree n must be a non-negative integer.
- 1Check n before applying factorial.
- 2A fractional polynomial degree is not valid here.
- 3The calculator returns an error.
Introduction
The Lagrange error bound estimates how far a Taylor polynomial can be from the true function value at a chosen point. Supply the derivative bound M, the degree n, the evaluation point x, and the expansion center a to get the remainder bound instantly.
What the Bound Measures
The Lagrange form of the remainder provides a guaranteed upper bound rather than the exact error.
M captures the largest possible magnitude of the next derivative on the interval.
The factorial term grows quickly and often makes higher-degree bounds smaller.
|x-a| measures how far you move from the expansion center.
The power term increases rapidly as that distance grows.
Required Inputs
Each input plays a distinct role in the bound.
M must be a positive finite number.
n must be a non-negative integer.
x is the point where the approximation is evaluated.
a is the center of the Taylor polynomial.
Why the Factorial Appears
The denominator comes from the structure of Taylor series coefficients.
(n+1)! is computed from the next derivative order.
For n = 3, the denominator is 4! = 24.
Larger n can produce very large factorials.
The calculator reports the factorial explicitly so you can verify the arithmetic.
Effect of Distance from the Center
The bound depends strongly on how far x is from a.
If x = a, then |x-a| = 0 and the bound is 0.
If |x-a| is small, the power term can shrink quickly.
If |x-a| is large, the error bound may become loose.
Moving closer to the center often improves approximation quality.
Worked Example
The sample values M = 2, n = 3, x = 1.5, a = 1 illustrate every intermediate term.
(n+1)! = 24.
|x-a| = 0.5.
|x-a|^(n+1) = 0.0625.
Final bound ≈ 0.005208.
Where This Is Used
Error bounds matter whenever approximation quality must be justified.
Taylor and Maclaurin series homework.
Numerical analysis and approximation theory.
Estimating truncation error in scientific computing.
Checking whether a chosen polynomial degree is sufficient.
FAQs
Is the result the exact error?
No. It is an upper bound on the possible error.
Why must n be an integer?
Taylor polynomial degree counts derivative orders, so it must be a whole number.
Can M be zero?
This version requires M > 0. If the next derivative is identically zero, the remainder is trivially zero.
What happens if x equals a?
Then the power term is zero and the bound is zero.
Why does the calculator show factorial and powerTerm?
Those intermediate values make the bound easier to verify by hand.
Does a smaller bound guarantee a better approximation?
Usually yes, but it still represents a worst-case estimate rather than the exact error.