Error Bounds

When we approximate a complicated function like sin(x), e^x, or ln(1+x) using a partial sum of its power series (called a Taylor or Maclaurin polynomial), we are cutting off an infinite sum after a finite number of terms. The pieces we throw away do not simply vanish — they contribute a leftover called the truncation error, or remainder. Understanding how large that remainder can be is essential whenever we need a guaranteed level of accuracy, whether programming a calculator, designing an aircraft navigation system, or proving a mathematical theorem. An error bound is a proven upper limit on the absolute value of the error. Rather than saying 'the error is probably small,' an error bound says 'the error is definitely no larger than some specific number.' The two most important tools for computing these bounds are the Alternating Series Estimation Theorem, which applies whenever the series alternates in sign and its terms shrink to zero, and the Lagrange Remainder Formula (also called Taylor's Remainder Theorem), which applies to any Taylor series and gives a bound in terms of a higher-order derivative. Mastering error bounds connects calculus to the real world of numerical computation. Every time a computer evaluates a trigonometric or exponential function, it is summing a polynomial approximation and relying on error analysis to know when to stop. By the end of this article you will be able to state and apply both major theorems, interpret their formulas term by term, and use them to determine how many terms of a series are needed to achieve a desired accuracy.

Key takeaways
  • What is the Remainder of a Power Series?
  • The Alternating Series Estimation Theorem
  • Taylor's Remainder Theorem (Lagrange Remainder)
  • Choosing the Right Theorem
  • Determining How Many Terms You Need

What is the Remainder of a Power Series?

If f(x) is represented by its Taylor series centered at a, the n-th degree Taylor polynomial is P_n(x) = sum_{k=0}^{n} f^(k)(a)/k! · (x-a)^k. The remainder (error) after using P_n is defined as R_n(x) = f(x) - P_n(x). Our goal is to bound |R_n(x)|, i.e., find a number M such that |R_n(x)| ≤ M for all x in a given interval.

The Alternating Series Estimation Theorem

Suppose a series satisfies the conditions of the Alternating Series Test: the terms alternate in sign, |a_{n+1}| ≤ |a_n| for all n, and lim_{n→∞} a_n = 0. Then the error made by stopping at the n-th partial sum S_n satisfies |R_n| = |S - S_n| ≤ |a_{n+1}|. In words: the absolute error is at most the absolute value of the FIRST OMITTED TERM. This is remarkably easy to apply — you do not need to know the actual sum S at all.

Taylor's Remainder Theorem (Lagrange Remainder)

For any function f whose (n+1)-th derivative exists, the remainder R_n(x) = f(x) - P_n(x) satisfies the Lagrange form: R_n(x) = f^(n+1)(c) / (n+1)! · (x-a)^(n+1) for some c strictly between a and x. Since we usually cannot find c exactly, we bound it: if |f^(n+1)(t)| ≤ M for all t between a and x, then |R_n(x)| ≤ M · |x-a|^(n+1) / (n+1)!. This bound works for any Taylor series, alternating or not.

Choosing the Right Theorem

Use the Alternating Series Estimation Theorem when the series naturally alternates (e.g., sin x, cos x, ln(1+x) near x=0) and the terms are decreasing — it gives the tightest bound with the least work. Use the Lagrange Remainder when the series does not alternate, or when you need a bound that works for ALL x in an interval, not just one specific value. For e^x (all positive terms), Lagrange is necessary.

Determining How Many Terms You Need

A common exam question: 'How many terms of the series for sin(0.1) are needed to guarantee accuracy within 0.0001?' Set your error bound less than or equal to 0.0001 and solve for n. For alternating series, solve |a_{n+1}| ≤ 0.0001. For Lagrange, find the smallest n such that M·|x-a|^(n+1)/(n+1)! ≤ 0.0001. Because factorials grow very fast, only a few terms are typically needed for small |x-a|.

The Key Equation

∣Rn(x)∣≤M⋅∣x−a∣n+1(n+1)!|R_n(x)| \leq \frac{M \cdot |x - a|^{n+1}}{(n+1)!}
  • R_n(x) — Remainder (Error) (same units as f(x)): The exact difference f(x) - P_n(x); what the truncated polynomial misses.
  • M — Derivative Bound (same units as f^(n+1)): An upper bound on |f^(n+1)(t)| for all t between a and x; found by analyzing the (n+1)-th derivative.
  • |x - a| — Distance from Center (same units as x): How far the evaluation point x is from the center a of the Taylor series.
  • n+1 — Next Degree (dimensionless): One more than the degree of the polynomial used; the error involves the (n+1)-th power and factorial.
  • (n+1)! — Factorial (dimensionless): The product 1 × 2 × 3 × ⋯ × (n+1); grows very rapidly, making the bound small for reasonable n.

Worked Example

Problem. Use the Maclaurin series for cos(x) to approximate cos(0.2). Then apply both the Alternating Series Estimation Theorem and the Lagrange Remainder to bound the error when using only the first two non-zero terms. Confirm the bounds are consistent.

cos⁡(0.2)≈0.9800667,∣R∣≤8.89×10−8\cos(0.2) \approx 0.9800667, \quad |R| \leq 8.89 \times 10^{-8}

Answer. cos(0.2) ≈ 0.9800667, with error at most 8.89 × 10⁻⁸ (alternating series) or 5.33 × 10⁻⁷ (Lagrange)

Figure
Diagram
Taylor Polynomial Approximations to cos(x)xy01-12-210.5error = R_n(x)at x=1cos(x) exactP₂(x)=1-x²/2P₄(x)=1-x²/2+x⁴/24
Origins · 1 minJoseph-Louis Lagrange and how this was discovered
JL
Joseph-Louis Lagrange
1736–1813 · Italian-French

Lagrange gave the remainder term in Taylor's theorem its explicit, computable form — now universally called the Lagrange Remainder. His 1797 work Théorie des fonctions analytiques attempted to build all of calculus on power series, making precise control of remainder terms central to mathematical analysis. The formula |R_n(x)| ≤ M|x-a|^{n+1}/(n+1)! that students use today is a direct descendant of his ideas. Lagrange was appointed chair of mathematics at the Royal Artillery School in Turin at age 19 — a position normally reserved for much older mathematicians. Napoleon so admired him that he called Lagrange 'the lofty pyramid of mathematical sciences.'

  • Taylor Series — Error bounds measure the accuracy of Taylor polynomial approximations; you must understand the Taylor series construction before applying remainder theorems.
  • Alternating Series Test — The Alternating Series Estimation Theorem — one of the two main error bound tools — only applies to series that satisfy the conditions of the Alternating Series Test.
  • Radius of Convergence — Error bounds are only valid within the interval of convergence; the radius of convergence determines where the power series (and its remainder formula) are meaningful.
  • Higher-Order Derivatives — The Lagrange Remainder depends on the (n+1)-th derivative of f, so computing and bounding higher-order derivatives is a prerequisite skill.