When you stop adding terms, how wrong are you? Two tools answer that — the simple
alternating-series bound, and the Lagrange bound for Taylor polynomials. This is the back half of
the unit, and the part students lose the most points on.
Every approximation cuts the series off somewhere, leaving a leftover — the remainder. You
rarely know it exactly, but you can bound it: prove the error is no bigger than some number.
That's what lets you say "this estimate is good to three decimal places" with certainty. Two cases
matter on the exam: an alternating series (easy) and a Taylor polynomial (Lagrange — the hard one).
Which bound do I use?
Half the battle is picking the right tool. There are exactly two error bounds in this
unit, and which one applies is decided by what you're approximating. Here's the choice, side by side.
Use this when…
you're truncating an alternating series (terms flip sign and shrink to 0)
How to spot it The series has a (−1)ⁿ factor, the terms get steadily smaller, and you're asked to "estimate" or "approximate" by stopping after some number of terms.
Quick example Approximating ln(1.5) with the series x − x²⁄2 + x³⁄3 − … at x = 0.5, stopping after 4 terms. The bound is just the 5th term: |0.5⁵⁄5| ≈ 0.0063. Done — no other work.
The bound is
the size of the first term you dropped, |an+1|
Difficulty
easy — a gift
Common trap Using it on a series that isn't actually alternating, or whose terms don't decrease — then the bound doesn't hold. Check the conditions first.
Use this when…
you're approximating a function with a Taylor / Maclaurin polynomial of degree n
How to spot it You're told to use a degree-n polynomial (or the first few terms) to estimate f at some point, and bound the error of that approximation.
Quick example Approximating e0.2 with a degree-3 Taylor polynomial. Go one past: M = max of the 4th derivative on [0, 0.2] = e0.2, and the bound is M·0.2⁴⁄4! ≈ 8.1×10⁻⁵.
The bound is
M·|x−c|n+1 ⁄ (n+1)! — one term past your polynomial
Difficulty
harder — four moving parts
Common trap Using degree n instead of n+1. Everything steps up by one: the (n+1)th derivative, the (n+1) power, and (n+1)!. Also: M is the max of the derivative, not its value at one point.
Both answer the same question — "if I stop here, how wrong can I be?" — and both give a
guarantee, not the exact error: a number your true error is promised not to exceed. You never
need the true value to compute either one. The rest of this page breaks down each method, then lets
you practice.
Method 1 — Alternating Series Error Bound
The easiest bound in the unit, and a gift when you can use it.
Method 2 — Lagrange Error Bound
The one everyone dreads — not because it's conceptually hard, but because the
formula has four moving parts and it's easy to plug something into the wrong slot. Here's each piece.
Practice
Work a full bound, single attempt per part, with the full reasoning shown when you
miss. Switch between the two bounds.