The Central Idea
Suppose an infinite process has exact value \(S\), while a finite approximation gives \(S_N\).
The error is
\[ \boxed{R_N=S-S_N}. \]Most practical estimation problems ask us to control the size
\[ |R_N|. \]What Is a Remainder?
A remainder is the part of an infinite process not yet included in a finite approximation.
For a convergent series,
\[ S=\sum_{n=1}^{\infty}a_n, \]the remainder after \(N\) terms is
\[ R_N=\sum_{n=N+1}^{\infty}a_n. \]Exact Error versus Error Bound
Sometimes \(R_N\) can be computed exactly.
More often, we find a number \(B_N\) such that
\[ \boxed{|R_N|\le B_N}. \]Then \(B_N\) is an error bound.
Absolute Error
The absolute error of an approximation \(A\) to an exact value \(S\) is
\[ \boxed{|S-A|}. \]It measures error in the same units as the quantity being approximated.
Relative Error
When \(S\ne0\), the relative error is
\[ \boxed{ \frac{|S-A|}{|S|} }. \]Relative error measures the size of the error compared with the size of the exact quantity.
Percent Error
Relative error may be expressed as a percentage:
\[ \boxed{ 100\frac{|S-A|}{|S|}\% }. \]Tolerance
A tolerance is a maximum acceptable error.
If the required tolerance is \(\varepsilon>0\), we want
\[ \boxed{|R_N|<\varepsilon}. \]Two Common Directions of a Problem
Error problems usually move in one of two directions:
- Given \(N\), estimate the error.
- Given a tolerance \(\varepsilon\), find an \(N\) large enough to guarantee it.
Convergence Does Not Give a Practical Error Bound by Itself
Knowing
\[ S_N\to S \]tells us that the error eventually becomes small.
It does not automatically tell us how large \(N\) must be for a particular tolerance.
Alternating-Series Remainder Estimate
Suppose
\[ \sum_{n=1}^{\infty}(-1)^{n-1}b_n \]satisfies the Alternating Series Test.
Then
\[ \boxed{|R_N|\le b_{N+1}}. \]The First Omitted Term
For a convergent alternating series with decreasing magnitudes, the first omitted term controls the entire remaining error.
This is one of the simplest useful error bounds in analysis.
Example: Alternating Harmonic Error
For
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n, \]after \(N\) terms,
\[ \boxed{|R_N|\le\frac1{N+1}}. \]Example: Ten-Term Alternating Harmonic Approximation
After \(10\) terms,
\[ |R_{10}| \le \frac1{11}. \]Therefore the approximation error is at most about
\[ 0.0909. \]Choosing Terms for an Alternating Series
To guarantee
\[ |R_N|<\varepsilon, \]choose \(N\) so that
\[ \boxed{b_{N+1}<\varepsilon}. \]Example: Alternating \(1/n^2\) Series
For
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^2}, \]require
\[ \frac1{(N+1)^2}<10^{-4}. \]Thus
\[ N+1>100, \]so \(N=100\) terms are sufficient.
Alternating Error Also Gives Direction
In a standard alternating series, the true sum lies between consecutive partial sums.
Therefore the next omitted term indicates whether the current approximation lies above or below the exact sum.
Two-Sided Alternating Bound
If \(S_N\) and \(S_{N+1}\) are consecutive partial sums,
\[ \boxed{ \min(S_N,S_{N+1}) \le S\le \max(S_N,S_{N+1}) }. \]Geometric-Series Remainder
For a convergent geometric series
\[ \sum_{n=0}^{\infty}ar^n, \qquad |r|<1, \]the exact sum is
\[ S=\frac{a}{1-r}. \]Finite Geometric Partial Sum
The partial sum through \(n=N\) is
\[ S_N = a\frac{1-r^{N+1}}{1-r}. \]Exact Geometric Remainder
Subtracting the partial sum from the exact sum gives
\[ \boxed{ R_N = \frac{ar^{N+1}}{1-r} }. \]Geometric Absolute Error
Therefore,
\[ \boxed{ |R_N| = \frac{|a||r|^{N+1}}{|1-r|} }. \]Example: Geometric Error
Consider
\[ \sum_{n=0}^{\infty}\left(\frac12\right)^n. \]After \(N\) terms through index \(N\),
\[ |R_N| = \frac{(1/2)^{N+1}}{1-1/2} = \left(\frac12\right)^N. \]Working Backward from a Geometric Tolerance
To require
\[ \frac{|a||r|^{N+1}}{|1-r|} <\varepsilon, \]solve the resulting exponential inequality for \(N\).
Logarithms Help Solve for \(N\)
When
\[ |r|^{N+1}Since \(0<|r|<1\), remember that \(\ln|r|<0\), so inequality direction must be handled carefully when dividing.
Example: Required Geometric Terms
For
\[ \sum_{n=0}^{\infty}\left(\frac13\right)^n, \]require error below \(10^{-5}\):
\[ \frac{(1/3)^{N+1}}{1-1/3}<10^{-5}. \]Solving this inequality determines a sufficient \(N\).
Taylor Approximation Error
If \(T_N(x)\) is the degree-\(N\) Taylor polynomial for \(f\) centered at \(a\), define
\[ \boxed{ R_N(x)=f(x)-T_N(x) }. \]The Taylor Remainder Measures What the Polynomial Misses
The polynomial matches derivatives through order \(N\) at the center.
The remainder measures the cumulative effect of all higher-order behavior not included in that finite polynomial.
Lagrange Form of the Taylor Remainder
Under the appropriate differentiability conditions, there exists some point \(\xi\) between \(a\) and \(x\) such that
\[ \boxed{ R_N(x) = \frac{f^{(N+1)}(\xi)}{(N+1)!} (x-a)^{N+1} }. \]Lagrange Error Bound
If
\[ |f^{(N+1)}(t)|\le M \]for every \(t\) between \(a\) and \(x\), then
\[ \boxed{ |R_N(x)| \le \frac{M|x-a|^{N+1}}{(N+1)!} }. \]The Three Ingredients in a Taylor Error Bound
The Lagrange bound depends on:
- a bound \(M\) for the next derivative;
- the distance \(|x-a|\) from the center;
- the factorial \((N+1)!\).
Distance from the Center Matters
The factor
\[ |x-a|^{N+1} \]explains why Taylor approximations are generally strongest near their center.
Factorial Growth Matters
The denominator
\[ (N+1)! \]grows very rapidly.
For many common functions, this causes Taylor errors to decrease quickly as \(N\) increases.
Example: Error for \(e^x\)
For
\[ f(x)=e^x, \]every derivative is \(e^x\).
On an interval between \(0\) and a positive \(x\), we can take
\[ M=e^x. \]Maclaurin Error Bound for \(e^x\)
The degree-\(N\) Maclaurin approximation satisfies
\[ |R_N(x)| \le \frac{M|x|^{N+1}}{(N+1)!}, \]where \(M\) bounds \(e^t\) between \(0\) and \(x\).
Example: Approximate \(e^{0.2}\)
Using the quadratic Maclaurin polynomial,
\[ T_2(x)=1+x+\frac{x^2}{2}. \]At \(x=0.2\),
\[ |R_2(0.2)| \le \frac{e^{0.2}(0.2)^3}{3!}. \]Sine and Cosine Have Convenient Derivative Bounds
Every derivative of \(\sin x\) or \(\cos x\) is one of
\[ \pm\sin x,\qquad \pm\cos x. \]Therefore,
\[ \boxed{|f^{(k)}(x)|\le1}. \]Taylor Error for Sine and Cosine
For Maclaurin approximations of \(\sin x\) or \(\cos x\), we can usually take
\[ M=1. \]Thus,
\[ |R_N(x)| \le \frac{|x|^{N+1}}{(N+1)!}. \]Example: Approximate \(\sin(0.1)\)
Use the linear approximation
\[ T_1(x)=x. \]A Lagrange bound gives
\[ |R_1(0.1)| \le \frac{(0.1)^2}{2!}. \]A Sharper Bound May Be Available
The general Lagrange bound is often convenient rather than optimal.
If additional structure is known, such as alternating behavior or vanishing coefficients, a tighter estimate may be possible.
Alternating Taylor Series
Some Taylor series are alternating for particular values of \(x\).
When the alternating-series hypotheses hold, the first omitted term may give a simpler or sharper error bound than the general Lagrange estimate.
Example: Cosine Near Zero
For positive \(x\) small enough,
\[ \cos x = 1-\frac{x^2}{2!} +\frac{x^4}{4!} -\cdots. \]If term magnitudes decrease, truncation error is bounded by the first omitted term.
First Omitted Taylor Term as an Alternating Bound
If we approximate
\[ \cos x \approx 1-\frac{x^2}{2}, \]then under the alternating-series conditions,
\[ |R| \le \frac{|x|^4}{4!}. \]Error Bounds Are Guarantees, Not Predictions
An error bound may overestimate the actual error.
The statement
\[ |R_N|\le B \]guarantees that the error is no larger than \(B\); it does not claim the error is exactly \(B\).
Actual Error Can Be Much Smaller
Conservative derivative bounds, cancellation, and special structure can all make the true error substantially smaller than the guaranteed upper bound.
Choosing \(M\) Carefully
In the Lagrange bound, \(M\) should satisfy
\[ |f^{(N+1)}(t)|\le M \]throughout the entire interval between the center and the evaluation point.
Do Not Evaluate the Derivative Only at the Center
The Lagrange remainder uses an unknown intermediate point \(\xi\).
Therefore a valid bound must control the derivative throughout the relevant interval, not merely at \(a\).
Example: Bounding \(e^x\) on an Interval
If \(0\le t\le0.5\), then
\[ e^t\le e^{0.5}. \]So \(M=e^{0.5}\) is a valid derivative bound for any Taylor remainder involving \(e^t\) on that interval.
Example: Bounding Rational Derivatives
For a function such as
\[ f(x)=\frac1{1-x}, \]derivatives grow as powers of \(1-x\) in the denominator.
A useful \(M\) requires identifying where the denominator is smallest on the interval under consideration.
Working Backward from Taylor Tolerance
To guarantee
\[ |R_N(x)|<\varepsilon, \]find \(N\) such that
\[ \frac{M|x-a|^{N+1}}{(N+1)!} <\varepsilon. \]There May Be No Simple Closed-Form Solution for \(N\)
Because factorials are involved, it is often fastest to test successive values of \(N\) until the inequality is satisfied.
A Practical Term-Selection Strategy
- Write the appropriate error bound.
- Insert the target \(x\) and center \(a\).
- Choose a valid derivative bound \(M\).
- Insert the tolerance \(\varepsilon\).
- Test increasing values of \(N\).
- Choose the smallest \(N\) that guarantees the required error.
Decimal Accuracy and Tolerance
To guarantee \(d\) decimal places by rounding, a common sufficient condition is
\[ |R_N|<\frac12\,10^{-d}. \]This places the approximation within half a unit in the \(d\)-th decimal place.
Example: Four Decimal Places
To guarantee correct rounding to four decimal places, it is sufficient to require
\[ |R_N|<0.00005. \]Error in a Series Sum versus Error in a Function Approximation
In a numerical series,
\[ R_N=S-S_N. \]In Taylor approximation,
\[ R_N(x)=f(x)-T_N(x). \]The same conceptual structure appears in both settings.
Tail Estimates for Positive-Term Series
Some positive-term convergence tests also provide quantitative remainder bounds.
The integral test is especially useful for this purpose.
Integral-Test Remainder Estimate
Suppose \(a_n=f(n)\), where \(f\) is positive, continuous, and decreasing, and
\[ \sum_{n=1}^{\infty}a_n \]converges.
Then
\[ \boxed{ \int_{N+1}^{\infty}f(x)\,dx \le R_N \le \int_N^{\infty}f(x)\,dx }. \]Why the Integral Estimate Works
For decreasing positive functions, rectangles representing series terms fit between neighboring areas under the curve.
The infinite tail can therefore be trapped between two improper integrals.
Example: Tail of a \(p\)-Series
For
\[ \sum_{n=1}^{\infty}\frac1{n^p}, \qquad p>1, \]the upper remainder estimate is
\[ R_N \le \int_N^\infty x^{-p}\,dx. \]Evaluating the \(p\)-Series Tail Bound
Since
\[ \int_N^\infty x^{-p}\,dx = \frac{N^{1-p}}{p-1}, \]we obtain
\[ \boxed{ R_N \le \frac{N^{1-p}}{p-1} }. \]Example: Tail of \(\sum1/n^2\)
For
\[ \sum_{n=1}^{\infty}\frac1{n^2}, \]the integral estimate gives
\[ R_N \le \int_N^\infty\frac1{x^2}\,dx = \frac1N. \]Integral Bounds Can Estimate the Sum Itself
Since
\[ S=S_N+R_N, \]the integral remainder bounds yield
\[ S_N+\int_{N+1}^{\infty}f(x)\,dx \le S \le S_N+\int_N^{\infty}f(x)\,dx. \]Bounding versus Approximating
An approximation gives a candidate numerical value.
A bound gives a guaranteed region containing the exact value.
Strong numerical work often uses both.
Intervals of Certainty
If
\[ |S-A|\le E, \]then
\[ \boxed{ A-E\le S\le A+E }. \]Example: Convert an Error Bound to an Interval
If
\[ A=1.4142 \]and
\[ |S-A|\le0.0001, \]then
\[ 1.4141\le S\le1.4143. \]Common Error: Confusing the Last Included Term with the First Omitted Term
If the partial sum ends at \(N\), the first omitted term is indexed by
\[ N+1. \]This matters in alternating and geometric remainder formulas.
Common Error: Treating an Error Bound as the Exact Error
If
\[ |R_N|\le0.001, \]the actual error may be \(0.001\), but it may also be much smaller.
Common Error: Forgetting Absolute Value
Error size is normally measured by
\[ |R_N|, \]not \(R_N\) alone.
Common Error: Using an Invalid \(M\)
A Lagrange bound fails if \(M\) does not actually bound the next derivative on the whole interval between \(a\) and \(x\).
Common Error: Using the Wrong Derivative Order
A degree-\(N\) Taylor polynomial uses the
\[ (N+1)\text{-st} \]derivative in the Lagrange remainder.
Common Error: Using \((N)!\) Instead of \((N+1)!\)
The Lagrange remainder is
\[ \frac{f^{(N+1)}(\xi)}{(N+1)!}(x-a)^{N+1}. \]Every index advances by one beyond the degree of the approximation.
Common Error: Ignoring the Distance from the Center
Taylor error depends on
\[ |x-a|^{N+1}. \]The same degree may give very different accuracy at different distances from the center.
Common Error: Assuming More Terms Always Give the Same Improvement
Error reduction depends on the structure of the series.
Geometric errors shrink by powers of \(|r|\), while Taylor errors often shrink through both powers and factorial growth.
Common Error: Rounding Too Early
Intermediate rounding can consume part of the allowed error budget.
Keep extra digits during calculation and round only at the end.
A Reliable Error-Estimation Procedure
- Identify the exact infinite object and the finite approximation.
- Define the remainder \(R_N\).
- Choose the remainder theorem appropriate to the structure.
- Write the bound symbolically before substituting numbers.
- Check all hypotheses.
- Evaluate the bound.
- State the conclusion with units or requested precision.
A Reliable Tolerance Procedure
- Write the target inequality \(|R_N|<\varepsilon\).
- Insert the appropriate remainder bound.
- Solve or test values for \(N\).
- Choose the smallest sufficient integer \(N\) when requested.
- Verify the final inequality.
Worked Example: Alternating Error
Estimate the error after \(20\) terms of
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3}. \]The first omitted magnitude is
\[ \frac1{21^3}. \]Therefore,
\[ \boxed{ |R_{20}| \le \frac1{9261} }. \]Worked Example: Geometric Error
Estimate the error after summing through \(n=5\) in
\[ \sum_{n=0}^{\infty}2\left(\frac14\right)^n. \]Here \(a=2\), \(r=1/4\), and \(N=5\):
\[ |R_5| = \frac{2(1/4)^6}{1-1/4}. \]Therefore,
\[ \boxed{ |R_5| = \frac1{1536} }. \]Worked Example: Taylor Error for \(e^x\)
Approximate \(e^{0.1}\) with
\[ T_2(x)=1+x+\frac{x^2}{2}. \]On \(0\le t\le0.1\),
\[ e^t\le e^{0.1}. \]Thus
\[ |R_2(0.1)| \le \frac{e^{0.1}(0.1)^3}{3!}. \]Worked Example: Taylor Error for Cosine
Approximate
\[ \cos(0.2) \]by
\[ 1-\frac{(0.2)^2}{2}. \]Using the alternating-series structure,
\[ |R| \le \frac{(0.2)^4}{4!}. \]Worked Example: Integral Tail Bound
Estimate the remainder after \(N\) terms of
\[ \sum_{n=1}^{\infty}\frac1{n^3}. \]Since
\[ \int_N^\infty x^{-3}\,dx = \frac1{2N^2}, \]we have
\[ \boxed{ R_N\le\frac1{2N^2} }. \]Practice 1: Alternating Error Bound
Find an upper bound for the error after \(12\) terms of
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^2}. \]Practice 2: Alternating Tolerance
How many terms of
\[ \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3} \]are sufficient to guarantee error less than \(10^{-4}\)?
Practice 3: Geometric Remainder
Find the exact remainder after summing through \(n=4\) in
\[ \sum_{n=0}^{\infty}\left(\frac12\right)^n. \]Practice 4: Geometric Tolerance
Find a sufficient \(N\) so that the remainder after summing through index \(N\) in
\[ \sum_{n=0}^{\infty}\left(\frac14\right)^n \]is less than \(10^{-5}\).
Practice 5: Lagrange Bound for \(e^x\)
Write a valid Lagrange error bound for approximating
\[ e^{0.3} \]by the degree-\(2\) Maclaurin polynomial.
Practice 6: Sine Error
Use a Taylor or alternating-series bound to estimate the error in
\[ \sin(0.1)\approx0.1. \]Practice 7: Cosine Error
Bound the error in
\[ \cos(0.3)\approx1-\frac{(0.3)^2}{2}. \]Practice 8: Integral-Test Tail
Find an upper bound for the remainder after \(N\) terms of
\[ \sum_{n=1}^{\infty}\frac1{n^4}. \]Practice 9: Convert a Bound to an Interval
An approximation gives
\[ A=2.7183 \]with
\[ |S-A|\le0.0002. \]Give an interval containing \(S\).
Practice 10: Identify the Error
A student uses a degree-\(4\) Taylor polynomial and writes
\[ |R_4(x)| \le \frac{M|x-a|^4}{4!}. \]Explain what is wrong.
Practice Answers
Practice 1
The first omitted term is \(n=13\):
\[ \boxed{ |R_{12}| \le \frac1{13^2} = \frac1{169} }. \]Practice 2
Require
\[ \frac1{(N+1)^3}<10^{-4}. \]Since
\[ 10^{4/3}\approx21.54, \]we need \(N+1>21.54\). Thus
\[ \boxed{N=21} \]terms are sufficient.
Practice 3
Here \(a=1\), \(r=1/2\), and \(N=4\):
\[ R_4 = \frac{(1/2)^5}{1-1/2} = \boxed{\frac1{16}}. \]Practice 4
The remainder is
\[ R_N = \frac{(1/4)^{N+1}}{1-1/4} = \frac43\left(\frac14\right)^{N+1}. \]Checking integers, \(N=8\) gives
\[ \frac43\left(\frac14\right)^9 <10^{-5}. \]Therefore
\[ \boxed{N=8} \]is sufficient.
Practice 5
For \(f(x)=e^x\), all derivatives equal \(e^x\). On \(0\le t\le0.3\),
\[ e^t\le e^{0.3}. \]Therefore,
\[ \boxed{ |R_2(0.3)| \le \frac{e^{0.3}(0.3)^3}{3!} }. \]Practice 6
Since
\[ \sin x=x-\frac{x^3}{3!}+\cdots, \]the first omitted term gives
\[ \boxed{ |R| \le \frac{(0.1)^3}{3!} }. \]Practice 7
Since
\[ \cos x = 1-\frac{x^2}{2!} +\frac{x^4}{4!}-\cdots, \]the first omitted term gives
\[ \boxed{ |R| \le \frac{(0.3)^4}{4!} }. \]Practice 8
Using the integral-test remainder estimate,
\[ R_N \le \int_N^\infty x^{-4}\,dx. \]Therefore,
\[ \boxed{ R_N \le \frac1{3N^3} }. \]Practice 9
Since
\[ |S-2.7183|\le0.0002, \]we have
\[ \boxed{ 2.7181\le S\le2.7185 }. \]Practice 10
A degree-\(4\) Taylor polynomial uses the fifth derivative and the fifth power in the Lagrange remainder:
\[ \boxed{ |R_4(x)| \le \frac{M|x-a|^5}{5!} }. \]What to Remember
The remainder is the difference between the exact infinite object and the finite approximation:
\[ \boxed{R_N=S-S_N} \]or, for Taylor approximation,
\[ \boxed{R_N(x)=f(x)-T_N(x)}. \]- alternating series: \(|R_N|\le b_{N+1}\);
- geometric series: \(\displaystyle R_N=\frac{ar^{N+1}}{1-r}\);
- integral-test tails: bound \(R_N\) with neighboring improper integrals;
- Taylor polynomials: \(\displaystyle |R_N(x)|\le\frac{M|x-a|^{N+1}}{(N+1)!}\).
The practical objective is not merely to know that an approximation improves, but to certify that its error is below a specified threshold.
Where We Go Next
Error estimates tell us how accurately finite polynomials and partial sums reproduce infinite processes. The next lesson uses that control to focus directly on representation and approximation of functions.
The next lesson, Function Approximation by Series, develops how series representations are selected, transformed, truncated, and used as practical computational models of functions.