Concept explainerCalculus IIAdvanced11 min read

Taylor remainder: how accurate is the polynomial?

A Taylor polynomial is useful only with a way to control what was left out.

|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}

Concept explainer

What the idea means, why its conditions matter, and where it connects.

Reviewed July 11, 2026

Error boundStart here

\boxed{|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}}

Choose M as an upper bound for the absolute value of the (n+1)st derivative between the center and x.

The polynomial and the remainder

The degree-n Taylor polynomial matches a function’s derivatives through order n at the center. The remainder R_n(x) is the exact difference between the function and that polynomial.

Matching many derivatives does not by itself guarantee global accuracy; distance from the center and derivative growth matter.

Using the Lagrange bound

Find a bound M for the next derivative on the interval connecting a and x. Insert M, the distance |x − a|, and the factorial denominator.

The estimate may be conservative, but it is rigorous and does not require knowing the unknown intermediate point in the exact remainder formula.

Alternating-series error can be sharper

When a Taylor series alternates with decreasing term magnitudes, the first omitted term bounds the error. Use that simpler result when its hypotheses are clearly satisfied.

Worked exampleBound an exponential approximation

Bound the error when e^0.2 is approximated by 1 + x + x²/2 at x = 0.2.

R_2(0.2)\le\frac{M(0.2)^3}{3!}

The next derivative order is three.

M=e^{0.2}

Every derivative of e^x is e^x, maximized at 0.2 on [0, 0.2].

|R_2(0.2)|\le\frac{e^{0.2}(0.008)}6

Insert the derivative bound and distance.

Result

\boxed{|R_2(0.2)|<0.00163}

Watch for

Common mistakes

Bounding the nth derivative instead of the (n+1)st.
Choosing M only at the center rather than across the whole interval.
Assuming a Taylor series equals the function without checking convergence or remainder.

Keep

Three takeaways

The remainder is the approximation error.
Bound the next derivative on the full interval.
Accuracy improves rapidly when factorial growth dominates.