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Remainders

Level 1 - Math I (Physics) topic page in Power Series.

Principle

A remainder measures the error made when an infinite series or a function is replaced by a finite approximation. It answers the practical question: how much could the omitted terms matter?

Approximation error
\[R_N(x)=f(x)-P_N(x)\]

Remainders are essential in numerical physics. They tell whether a truncated expansion is accurate enough for a calculation or experiment.

Notation

\(R_N(x)\)
remainder after keeping terms through degree N
\(P_N(x)\)
degree N Taylor polynomial
\(M\)
upper bound for a derivative or term size
\(\epsilon\)
allowed error tolerance
\(T_n\)
nth term of a series

The best remainder estimate depends on the type of series. Taylor's theorem, alternating-series estimates, and geometric tails are common Level 1 tools.

Method

Step 1: Identify the approximation

Decide what is being kept and what is being omitted. For a Taylor polynomial, the omitted part starts at degree \(N+1\).

Step 2: Choose an error estimate

For Taylor polynomials with bounded higher derivative, use a Lagrange-style bound. For a geometric tail, use the exact tail formula. For alternating decreasing terms, the next term bounds the error.

Step 3: Compare with tolerance

If an error tolerance \(\epsilon\) is required, solve an inequality such as

Start with an error bound
\[|R_N(x)|\le B_N\]
Require the error to be acceptable
\[B_N\le\epsilon\]
Solve for the degree
\[N\text{ large enough}\]

Rules

Taylor remainder bound
\[|R_N(x)|\le\frac{M}{(N+1)!}|x-c|^{N+1}\]
Geometric tail
\[\sum_{n=N+1}^{\infty}ar^n=\frac{ar^{N+1}}{1-r}\quad(|r|\lt1)\]
Alternating-series estimate
\[|R_N|\le |T_{N+1}|\]

The alternating estimate requires terms that decrease in size to zero and alternate in sign.

Examples

Question
Using
\[e^x\approx1+x\]
bound the error at
\[x=0.05\]
if
\[e^t\le2\]
on the interval.
Answer
The first omitted Taylor order is quadratic. Use
\[|R_1(x)|\le M|x|^2/2!\]
with
\[M=2\]
At
\[x=0.05\]
\[|R_1|\le2(0.05)^2/2=0.0025\]

Checks

  • State what has been kept before estimating what has been omitted.
  • A remainder is an error term, not another approximation by itself.
  • Geometric and alternating estimates have conditions; check them first.
  • In physics calculations, compare the remainder with the scale of the measured quantity.

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Taylor Theorem

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Taylor Limits