Lagrange Error Bound Calculator

What Is the Lagrange Error Bound?

A Taylor polynomial Pₙ(x) approximates a function f(x) near a center a. The difference f(x) − Pₙ(x) is called the remainder, Rₙ(x). Working it out exactly needs f(x) itself, which is what you were trying to avoid, so instead you put a ceiling on it.

The Lagrange error bound (also called the Taylor remainder bound or Taylor's inequality) is that ceiling. It guarantees that Pₙ(x) is within a known distance of the true value, which is how calculators and textbooks prove how many decimal places an approximation is good for. It is a standard topic in AP Calculus BC and first-year calculus.

The Lagrange Error Bound Formula

Taylor's theorem with the Lagrange form of the remainder says that for some number c between a and x:

You do not know c, but if |f⁽ⁿ⁺¹⁾(t)| ≤ M for every t between a and x, then:

• n is the degree of the Taylor polynomial.
• a is the center, and x is the point where you use the approximation.
• M bounds the next derivative, the (n + 1)th, between a and x.
• (n + 1)! grows very fast, which is why more terms usually mean a much smaller error.

How to Find M

M is the largest value of |f⁽ⁿ⁺¹⁾(t)| on the interval between a and x. Any larger number also works, but gives a weaker bound. For the functions in the calculator:

For other functions, find the (n + 1)th derivative, then its largest absolute value on the interval: check the endpoints and any turning points, or use a simple overestimate. Many textbooks use M = 3 for eˣ on [0, 1], since e < 3.

Bound vs Actual Error

The bound is a guarantee, not a prediction. Here is sin 1 approximated by Taylor polynomials centered at 0, with M = 1:

The actual error is always below the bound. A useful trick: for sin x at 0, the x⁴ term is 0, so P₃ = P₄ and you can use the n = 4 bound 1/120 ≈ 0.0083 for P₃ as well, which is much closer to the real 0.0081.

Worked Examples

sin 1 with degree 4. Every derivative of sin x is ±sin or ±cos, so M = 1.

P₄(1) = 1 − 1/6 ≈ 0.8333333 and sin 1 ≈ 0.8414710, so the actual error is 0.0081377, inside the bound.

e^0.5 with degree 3. Every derivative is eᵗ, largest at t = 0.5, so M = e^0.5 ≈ 1.6487:

Using the rougher M = 2 gives 0.0052083, still a valid bound.

ln(1.2) with degree 2. The third derivative of ln(1 + x) is 2 ÷ (1 + t)³, largest at t = 0, so M = 2 and the bound is 2 × 0.2³ ÷ 3! ≈ 0.0026667. P₂(0.2) = 0.2 − 0.02 = 0.18, and ln 1.2 ≈ 0.1823216.

How Many Terms Do You Need?

Turn the bound around: find the smallest n that makes the bound smaller than the accuracy you need. To approximate e = e¹ with an error below 10⁻⁶, use M = 3:

So the degree 9 Taylor polynomial, 1 + 1 + 1/2! + … + 1/9!, gives e to within 10⁻⁶. Try different values of n in the calculator to find the cut-off quickly.

Lagrange vs Alternating Series Error Bound

When a Taylor series has terms that alternate in sign and shrink towards 0 (like sin 1 or ln 1.2), the alternating series error bound also applies: the error is at most the first term left out. For sin 1 with P₃, that term is 1/5! = 1/120, the same as the Lagrange bound with n = 4.

The Lagrange bound works for any Taylor polynomial, alternating or not (for example eˣ at x > 0, where every term is positive), so it is the one to use when the series does not alternate.

Common Mistakes

• Using n! instead of (n + 1)!. The bound uses the next factorial and the power n + 1.
• Bounding the wrong derivative. M bounds f⁽ⁿ⁺¹⁾, not f⁽ⁿ⁾.
• Taking M at the wrong end. For eˣ use the larger of a and x; for ln(1 + x) the smaller.
• Using the value at c. c is unknown; M must cover every t between a and x.
• Reading the bound as the error. The true error is usually smaller.
• Degrees vs radians. Taylor series for sin and cos need x in radians.

Key Terms

Pₙ(x)

The Taylor polynomial of degree n about a.

Rₙ(x)

The remainder f(x) − Pₙ(x), the true error.

Center

The point a the Taylor polynomial is built around.

M

An upper limit for |f⁽ⁿ⁺¹⁾(t)| between a and x.

Maclaurin

A Taylor polynomial centered at a = 0.

How to Use the Lagrange Error Bound Calculator

1. Pick the function. Choose sin x, cos x, eˣ, ln(1 + x) or 1/(1 − x), and the calculator works out M for you. Choose Other for any other function and type M yourself.
2. Enter n, a and x. Type the degree n of the Taylor polynomial (0 to 100), the center a and the point x. a and x can be decimals, fractions such as 1/2, or multiples of π such as π/6.
3. Check M. Leave the M field empty to use the worked-out value, or type your own M, the largest value of |f⁽ⁿ⁺¹⁾(t)| between a and x.
4. Read and copy the bound. The Lagrange error bound appears as you type. The table shows M, |x − a|, (n + 1)!, the Taylor polynomial value, f(x) and the actual error. Press Copy to copy the bound, or Clear or Esc to start again.

Example: pick sin x and enter n = 4, a = 0 and x = 1. The bound is 0.008333333333 (1/120), and the table shows P₄(1) = 0.8333333333, sin 1 = 0.8414709848 and the actual error 0.008137651475.

Who Can Use This Calculator?

Frequently Asked Questions (FAQs)