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The Lagrange error bound puts a hard, guaranteed ceiling on how far a Taylor polynomial of degree n can possibly be from the true function value at a given point — without needing to know the exact function value at all. This matters enormously in numerical computing and calculus: whenever a calculator, computer, or physics model approximates a complicated function (like sine, cosine, eˣ, or ln(x)) using a truncated polynomial for speed, the Lagrange error bound is what proves mathematically how many decimal places of accuracy that approximation can be trusted to.
How it works
Enter M (an upper bound on the absolute value of the function's (n+1)th derivative, over the entire interval between the center a and the target point x), the center of expansion a, the target point x, and the polynomial's degree n. The calculator applies the Lagrange error bound formula: |Rₙ(x)| ≤ M/(n+1)! × |x−a|ⁿ⁺¹. Intuitively, this formula comes directly out of the remainder term in Taylor's theorem — it says the "leftover" error after truncating a Taylor series at degree n behaves like the NEXT term in the series would (using the (n+1)th derivative and an (n+1)th power), just with a worst-case upper bound (M) substituted in place of the exact but generally unknown derivative value.
- Enter m — max value of the (n+1)th derivative on the interval.
- Enter center of expansion (a).
- Enter target point (x).
- Enter degree of the Taylor polynomial (n).
- Click Calculate to see your results.
Examples
M = 1, a = 0, x = 0.5, n = 3
Bound = 1/4! × |0.5−0|⁴ = 1/24 × 0.0625 ≈ 0.0026.
M = 2, a = 0, x = 1, n = 4
Bound = 2/5! × |1−0|⁵ = 2/120 × 1 ≈ 0.0167.
M = 0.5, a = 2, x = 2.2, n = 2
Bound = 0.5/3! × |2.2−2|³ = 0.5/6 × 0.008 ≈ 0.00067.
Who should use it
- Determining how many Taylor series terms are needed for a target accuracy.
- Calculus coursework on Taylor polynomials and series convergence.
- Validating numerical approximation error in scientific computing.
Industry applications
- Calculus education
- Numerical analysis and scientific computing
Advantages
- Gives a rigorous, provable worst-case guarantee rather than just an empirical estimate.
- Directly tied to the standard Taylor remainder theorem taught in calculus courses.
Limitations
- Requires knowing (or reasonably bounding) the (n+1)th derivative's maximum, which can be difficult for complicated functions.
- Can be a very loose (overly pessimistic) bound if M isn't tightly estimated.
Common mistakes to avoid
- Using n instead of n+1 in the factorial or the exponent — both must use n+1, one more than the polynomial's degree, not the degree itself.
- Choosing an M that only bounds the derivative AT a or AT x, rather than over the ENTIRE interval between them, which can understate the true worst-case error.
- Treating the computed bound as the exact error value rather than an upper limit that the true error might fall well short of.
Best practices
- Always bound the (n+1)th derivative over the FULL interval between a and x, not just at the endpoints.
- Choose the smallest valid M you can justify, since an unnecessarily large M produces an overly pessimistic (loose) error bound.
- Remember the bound is most useful and tightest for x close to the center a — expect looser bounds and slower convergence far from the center.
Tips
- Need the actual harmonic-series-style sum estimate instead? See the Harmonic Number Calculator.