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Taylor's Inequality — Definition, Formula & Examples

Taylor's Inequality is a formula that tells you the maximum possible error when you approximate a function using a Taylor polynomial of degree nn. It bounds the remainder term Rn(x)R_n(x) by relating it to the size of the (n+1)(n+1)th derivative on the interval between the center and the point of approximation.

If ∣f(n+1)(t)∣≤M|f^{(n+1)}(t)| \le M for all tt between aa and xx, then the remainder Rn(x)=f(x)−Tn(x)R_n(x) = f(x) - T_n(x) of the nnth-degree Taylor polynomial centered at aa satisfies ∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \le \dfrac{M}{(n+1)!}|x - a|^{n+1}.

Key Formula

∣Rn(x)∣≤M(n+1)! ∣x−a∣n+1|R_n(x)| \le \frac{M}{(n+1)!}\,|x - a|^{n+1}
Where:
  • Rn(x)R_n(x) = The remainder (error) between f(x) and its nth-degree Taylor polynomial
  • MM = An upper bound for |f^{(n+1)}(t)| on the interval between a and x
  • nn = The degree of the Taylor polynomial
  • aa = The center of the Taylor expansion
  • xx = The point at which you are approximating f

How It Works

To use Taylor's Inequality, first identify the degree nn of your Taylor polynomial and the center aa. Then find a bound MM on ∣f(n+1)(t)∣|f^{(n+1)}(t)| for all tt between aa and xx—this is often the trickiest step and may require knowing that the derivative is increasing or bounded by a convenient constant. Plug MM, nn, aa, and xx into the formula to get the maximum error. If the bound is not small enough, increase nn and repeat.

Worked Example

Problem: Estimate the error when approximating e0.1e^{0.1} using the 3rd-degree Taylor polynomial for exe^x centered at a=0a = 0.
Identify derivatives: Every derivative of exe^x is exe^x. We need a bound MM on ∣f(4)(t)∣=et|f^{(4)}(t)| = e^t for t∈[0,0.1]t \in [0, 0.1].
Find M: Since ete^t is increasing, its maximum on [0,0.1][0, 0.1] is e0.1e^{0.1}. We can safely overestimate: M=e0.1<e<3M = e^{0.1} < e < 3.
M=3M = 3
Apply the inequality: With n=3n = 3, a=0a = 0, and x=0.1x = 0.1:
∣R3(0.1)∣≤34!(0.1)4=324(0.0001)=0.0000125|R_3(0.1)| \le \frac{3}{4!}(0.1)^4 = \frac{3}{24}(0.0001) = 0.0000125
Answer: The error in the approximation is at most 1.25×10−51.25 \times 10^{-5}, meaning the 3rd-degree polynomial gives e0.1e^{0.1} accurate to about four decimal places.

Why It Matters

Taylor's Inequality is essential whenever you need a guaranteed precision for a numerical approximation—common in scientific computing, engineering calculations, and numerical analysis courses. It also provides one standard method for proving that a Taylor series converges to its function by showing Rn(x)→0R_n(x) \to 0 as n→∞n \to \infty.

Common Mistakes

Mistake: Using the value of the (n+1)(n+1)th derivative only at the center aa instead of finding its maximum on the entire interval [a,x][a, x].
Correction: You must bound ∣f(n+1)(t)∣|f^{(n+1)}(t)| for all tt between aa and xx. Evaluate or overestimate the derivative on the whole interval to get a valid MM.