The Central Idea
Suppose a function has a convergent series representation
\[ f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n. \]Replacing the infinite series by its first \(N+1\) terms gives
\[ \boxed{ P_N(x)=\sum_{n=0}^{N}c_n(x-a)^n }. \]The finite polynomial \(P_N\) is an approximation to \(f\).
Why Approximate a Function?
Many functions are easy to describe symbolically but expensive or inconvenient to evaluate directly.
A polynomial approximation can replace the original function with additions and multiplications.
Series Approximation Has Three Questions
- What series represents the function?
- Where does that representation converge?
- How many terms are needed for the desired accuracy?
A complete approximation addresses all three.
Representation Comes Before Truncation
Before using a partial sum as an approximation, establish or identify a valid infinite series representation.
The polynomial truncation inherits its meaning from that infinite series.
Partial Sums Are Polynomial Approximations
If
\[ f(x)=c_0+c_1(x-a)+c_2(x-a)^2+\cdots, \]then the first approximations are
\[ P_0(x)=c_0, \] \[ P_1(x)=c_0+c_1(x-a), \] \[ P_2(x)=c_0+c_1(x-a)+c_2(x-a)^2. \]Approximation Improves Through Refinement
Each additional term retains all previous terms and adds another correction:
\[ P_{N+1}(x)=P_N(x)+c_{N+1}(x-a)^{N+1}. \]This creates a layered approximation process.
The Remainder Measures the Missing Part
Define
\[ R_N(x)=f(x)-P_N(x). \]Then
\[ \boxed{ f(x)=P_N(x)+R_N(x) }. \]Approximation Is Useful Only Inside a Valid Region
A power series may represent a function only on its interval of convergence.
Outside that interval, adding more terms need not improve the approximation at all.
The Center Organizes the Approximation
In
\[ \sum c_n(x-a)^n, \]the center is \(a\).
Near \(a\), powers of \(x-a\) are small, which often makes low-degree approximations especially effective.
Choosing a Center Near the Evaluation Point
If several valid centers are available, choose one near the desired \(x\)-value.
Smaller
\[ |x-a| \]usually means faster decay of higher powers and better finite approximations.
Maclaurin Approximation
When the center is \(0\),
\[ f(x) = \sum_{n=0}^{\infty}c_nx^n. \]Maclaurin series are convenient when the target \(x\) is near zero or when the function has especially simple derivative values there.
Taylor Approximation
When coefficients come from derivatives at \(a\),
\[ c_n=\frac{f^{(n)}(a)}{n!}, \]the approximation is a Taylor polynomial:
\[ T_N(x) = \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n. \]The Exponential Series
\[ e^x = 1+x+\frac{x^2}{2!} +\frac{x^3}{3!} +\frac{x^4}{4!} +\cdots. \]This series converges for every real \(x\).
Approximating \(e^x\) with a Polynomial
The degree-\(3\) Maclaurin approximation is
\[ \boxed{ e^x \approx 1+x+\frac{x^2}{2}+\frac{x^3}{6} }. \]Example: Approximate \(e^{0.2}\)
Use
\[ e^{0.2} \approx 1+0.2+\frac{0.2^2}{2} +\frac{0.2^3}{6}. \]This gives
\[ e^{0.2}\approx1.221333\ldots. \]The Sine Series
\[ \sin x = x-\frac{x^3}{3!} +\frac{x^5}{5!} -\frac{x^7}{7!} +\cdots. \]Low-Degree Sine Approximation
Near zero,
\[ \boxed{ \sin x\approx x }. \]A more accurate cubic approximation is
\[ \boxed{ \sin x\approx x-\frac{x^3}{6} }. \]The Small-Angle Approximation
The familiar approximation
\[ \sin x\approx x \]is simply the first nonzero truncation of the Maclaurin series.
It assumes \(x\) is measured in radians.
The Cosine Series
\[ \cos x = 1-\frac{x^2}{2!} +\frac{x^4}{4!} -\frac{x^6}{6!} +\cdots. \]Low-Degree Cosine Approximation
Near zero,
\[ \boxed{ \cos x\approx1-\frac{x^2}{2} }. \]The Logarithmic Series
\[ \ln(1+x) = x-\frac{x^2}{2} +\frac{x^3}{3} -\frac{x^4}{4} +\cdots. \]Its real interval of convergence is
\[ -1Approximating \(\ln(1+x)\)
For small \(x\),
\[ \boxed{ \ln(1+x)\approx x }. \]Adding another term gives
\[ \boxed{ \ln(1+x)\approx x-\frac{x^2}{2} }. \]The Arctangent Series
\[ \arctan x = x-\frac{x^3}{3} +\frac{x^5}{5} -\frac{x^7}{7} +\cdots. \]This gives polynomial approximations to inverse tangent near zero.
The Geometric Series as a Representation Engine
The identity
\[ \frac1{1-x} = \sum_{n=0}^{\infty}x^n, \qquad |x|<1, \]can be transformed into many useful function series.
Approximate a Rational Function
For
\[ \frac1{1-x}, \]the degree-\(N\) approximation is
\[ \boxed{ 1+x+x^2+\cdots+x^N }. \]Example: Approximate \(1/(1-0.1)\)
Using four terms,
\[ \frac1{1-0.1} \approx 1+0.1+0.1^2+0.1^3. \]The approximation is
\[ 1.111. \]Transforming a Known Series
A practical series library becomes much more useful when combined with:
- substitution;
- multiplication by constants or powers;
- addition and subtraction;
- differentiation;
- integration.
Substitution
If
\[ f(u)=\sum c_nu^n, \]then replacing \(u\) by \(g(x)\) gives
\[ f(g(x)) = \sum c_n[g(x)]^n \]wherever the transformed series converges.
Example: Approximate \(e^{-x^2}\)
Substitute \(-x^2\) into the exponential series:
\[ e^{-x^2} = 1-x^2+\frac{x^4}{2!} -\frac{x^6}{3!} +\frac{x^8}{4!} -\cdots. \]Example: Approximate \(\sin(2x)\)
Substitute \(2x\) into the sine series:
\[ \sin(2x) = 2x-\frac{(2x)^3}{3!} +\frac{(2x)^5}{5!} -\cdots. \]Multiplying a Series by a Power
If
\[ e^x = 1+x+\frac{x^2}{2!}+\cdots, \]then
\[ x^2e^x = x^2+x^3+\frac{x^4}{2!} +\frac{x^5}{3!}+\cdots. \]Adding Series
If two power series converge in a common region, they can be added term by term.
This can construct approximations for sums of functions without recomputing all derivatives from scratch.
Subtracting Series
Subtracting known expansions can expose cancellation and simplify the leading behavior of a function.
Example: Approximate \(e^x-1\)
\[ e^x-1 = x+\frac{x^2}{2!} +\frac{x^3}{3!} +\cdots. \]Near zero, the leading approximation is
\[ \boxed{e^x-1\approx x}. \]Example: Approximate \(1-\cos x\)
\[ 1-\cos x = \frac{x^2}{2!} -\frac{x^4}{4!} +\frac{x^6}{6!} -\cdots. \]Therefore,
\[ \boxed{ 1-\cos x\approx\frac{x^2}{2} }. \]Leading-Term Approximation
Near the center, the first nonzero term often gives the dominant local behavior.
Examples include
\[ \sin x\sim x, \] \[ e^x-1\sim x, \] \[ 1-\cos x\sim\frac{x^2}{2} \]as \(x\to0\).
Approximation and Asymptotic Equivalence
Writing
\[ f(x)\sim g(x) \quad\text{as }x\to a \]means
\[ \frac{f(x)}{g(x)}\to1. \]Series expansions often reveal such relationships immediately.
Why Leading Terms Matter
Leading terms can:
- simplify limits;
- estimate small corrections;
- compare growth rates;
- build local models;
- guide numerical algorithms.
Polynomial Degree Controls Model Complexity
A low-degree approximation is inexpensive but less detailed.
A higher-degree approximation includes more local information but requires more computation.
Accuracy versus Cost
Practical approximation balances:
- desired accuracy;
- number of terms;
- evaluation cost;
- distance from the center;
- available error bounds.
More Terms Are Not the Only Strategy
If convergence is slow at the desired point, it may be more effective to:
- choose a closer center;
- rewrite the function;
- transform the argument;
- use a different known series.
Example: Shift the Center for \(e^x\)
To approximate \(e^{2.01}\), a Maclaurin series uses powers of \(2.01\).
A Taylor series centered at \(2\) uses
\[ x-2=0.01, \]which is far smaller.
Factor Out a Convenient Known Value
Since
\[ e^{2.01}=e^2e^{0.01}, \]we can approximate only the small exponential correction:
\[ e^{0.01} = 1+0.01+\frac{0.01^2}{2!}+\cdots. \]Argument Reduction
Rewriting a function so that a series is evaluated at a smaller argument can dramatically improve convergence.
This principle appears throughout numerical computation.
Series Approximation for Definite Integrals
If a function is represented by a power series, we can often integrate the polynomial truncation term by term.
This provides approximations to integrals that may not have elementary antiderivatives.
Example: Integrating \(e^{-x^2}\)
Since
\[ e^{-x^2} = 1-x^2+\frac{x^4}{2!} -\frac{x^6}{3!}+\cdots, \]integrating from \(0\) to \(b\) gives
\[ \int_0^b e^{-x^2}\,dx = b-\frac{b^3}{3} +\frac{b^5}{5\cdot2!} -\frac{b^7}{7\cdot3!} +\cdots. \]Series Approximation for Derivatives
A power series can also be differentiated term by term inside its radius of convergence.
This produces polynomial approximations for derivatives as well as for the original function.
Example: Derivative from an Approximation
If
\[ \sin x \approx x-\frac{x^3}{6}+\frac{x^5}{120}, \]then differentiating gives
\[ \cos x \approx 1-\frac{x^2}{2}+\frac{x^4}{24}. \]Approximation of Solutions to Equations
Series expansions can replace complicated function equations with polynomial equations near a chosen point.
The polynomial model can then provide an initial estimate for a root or other feature.
Approximation in Modeling
In applied mathematics, a complicated relationship may be replaced locally by its first few series terms.
This is especially useful when a variable represents a small perturbation around a known operating point.
Zero-Order Approximation
The simplest local model is
\[ f(x)\approx f(a). \]This treats the function as approximately constant near \(a\).
First-Order Approximation
Adding slope information gives
\[ \boxed{ f(x)\approx f(a)+f'(a)(x-a) }. \]This is linearization.
Second-Order Approximation
Adding curvature gives
\[ \boxed{ f(x)\approx f(a)+f'(a)(x-a) +\frac{f''(a)}{2}(x-a)^2 }. \]Order Means the Highest Retained Power
A first-order approximation keeps terms through \((x-a)^1\).
A second-order approximation keeps terms through \((x-a)^2\), and so on.
Higher Order Is Not Automatically Better Everywhere
Within a valid convergent series, increasing \(N\) eventually improves the approximation at a fixed point.
But a high-degree polynomial evaluated outside the useful region can behave very poorly.
Local Approximation versus Global Behavior
Taylor information is centered at one point.
A polynomial may match a function extremely well near the center while differing substantially farther away.
The Radius of Convergence Is a Structural Boundary
For a power series,
\[ |x-a|Approximation strategy must respect that boundary.
Near the Boundary, Convergence May Slow
Even inside the interval of convergence, a series may require many more terms near an endpoint than near its center.
Example: Geometric Convergence Near the Boundary
In
\[ \frac1{1-x} = 1+x+x^2+\cdots, \]the terms decay rapidly when \(x=0.1\), but slowly when \(x=0.9\).
Both values lie inside \(|x|<1\), but their computational behavior is very different.
Error Control Completes the Approximation
A statement such as
\[ f(x)\approx P_N(x) \]becomes quantitatively meaningful when paired with
\[ \boxed{|f(x)-P_N(x)|\le E}. \]Use the Best Available Remainder Tool
Depending on the series, useful tools include:
- the first omitted term for alternating series;
- the exact geometric remainder;
- the Lagrange remainder for Taylor polynomials;
- integral tail estimates for suitable positive series.
Example: Approximation with an Error Guarantee
If
\[ \cos x = 1-\frac{x^2}{2!} +\frac{x^4}{4!}-\cdots \]has decreasing alternating term magnitudes at the chosen \(x\), then
\[ \cos x \approx 1-\frac{x^2}{2} \]with
\[ |R| \le \frac{|x|^4}{4!}. \]Horner Form for Efficient Evaluation
A polynomial such as
\[ c_0+c_1u+c_2u^2+c_3u^3 \]can be rewritten as
\[ \boxed{ c_0+u(c_1+u(c_2+uc_3)) }. \]This is called Horner form.
Why Horner Form Helps
Horner form reduces the number of multiplications needed to evaluate a polynomial and avoids computing each power separately.
It is a standard numerical implementation strategy for polynomial approximations.
Example: Horner Evaluation
The cubic exponential approximation
\[ 1+x+\frac{x^2}{2}+\frac{x^3}{6} \]can be written
\[ \boxed{ 1+x\left(1+x\left(\frac12+\frac{x}{6}\right)\right) }. \]Recursive Term Evaluation
Many series terms can be generated from the previous term rather than recomputed from scratch.
For the exponential series,
\[ t_n=\frac{x^n}{n!}, \]and
\[ \boxed{ t_{n+1}=t_n\frac{x}{n+1} }. \]Why Recursive Terms Help
Recursive evaluation:
- avoids repeated factorial calculations;
- avoids repeated exponentiation;
- supports efficient numerical summation;
- makes stopping rules easier to implement.
Stopping Rules
In numerical work, terms are often added until a mathematically justified error condition is satisfied.
A small term alone is not always enough; the stopping rule should match the available remainder theorem.
Do Not Confuse a Small Next Term with a Universal Error Bound
The first omitted term bounds the error for certain alternating series.
It is not a universal remainder rule for arbitrary power series.
Round Only After the Approximation Is Stable
Repeated rounding during a series calculation can introduce additional numerical error.
Keep extra digits until the final result.
Truncation Error and Rounding Error
Truncation error comes from stopping an infinite process.
Rounding error comes from finite numerical precision.
Good numerical practice manages both.
Approximation Can Be Symbolic or Numerical
A truncated series may be kept as a polynomial formula,
\[ P_N(x), \]or evaluated at a specific numerical input.
The symbolic form can be reused at many nearby inputs.
Reusable Local Models
If many evaluations are needed near the same center, constructing one Taylor polynomial can be more efficient than repeatedly evaluating the original function.
Approximation and Sensitivity
The first derivative term shows first-order sensitivity:
\[ \Delta f\approx f'(a)\Delta x. \]Higher-order terms describe nonlinear corrections to that sensitivity.
Approximation and Scale
Series terms often naturally rank corrections by powers of a small quantity.
This creates a hierarchy:
\[ 1,\quad \varepsilon,\quad \varepsilon^2,\quad \varepsilon^3,\ldots \]when \(|\varepsilon|\ll1\).
Neglecting Higher-Order Terms
If \(|\varepsilon|\) is small, then
\[ |\varepsilon|^2\ll|\varepsilon| \]and
\[ |\varepsilon|^3\ll|\varepsilon|^2. \]This explains why low-order approximations are often effective in perturbative settings.
Big-O Notation as an Approximation Summary
A local expansion may be summarized as
\[ f(x)=P_N(x)+O((x-a)^{N+1}) \]when the neglected remainder is controlled at that order near \(a\).
Example: Local Big-O Form
Near \(x=0\),
\[ \sin x = x-\frac{x^3}{6}+O(x^5). \]This indicates that the next scale of neglected behavior is of order \(x^5\).
Big-O Is Not an Exact Numerical Error Bound
Big-O notation describes asymptotic order.
A specific numerical tolerance still requires an explicit bound or additional constant information.
Common Error: Approximating Outside the Convergence Interval
A power series approximation must be used where its infinite series actually converges.
Common Error: Choosing Too Few Terms Without Checking Error
A visually simple polynomial is not automatically accurate enough.
Use a remainder estimate when accuracy matters.
Common Error: Assuming More Terms Fix a Bad Center
If the evaluation point is near a convergence boundary, convergence may remain slow.
Re-centering or transforming the problem may be more efficient.
Common Error: Ignoring the Units of the Input
Trigonometric Taylor series require radian measure.
Using degrees directly changes the numerical meaning of the expansion.
Common Error: Dropping the Center Shift
A series centered at \(a\) uses powers of
\[ x-a. \]Replacing them by powers of \(x\) creates a different approximation.
Common Error: Using a Series Identity Beyond Its Domain
An algebraically correct series formula may have a restricted convergence interval.
Always carry the convergence condition through substitutions and transformations.
Common Error: Confusing Approximate Equality with Exact Equality
A truncated polynomial satisfies
\[ f(x)\approx P_N(x), \]not generally
\[ f(x)=P_N(x). \]Common Error: Forgetting the Remainder
The exact relationship is
\[ f(x)=P_N(x)+R_N(x). \]The remainder is the mathematical record of what truncation removes.
A Reliable Function-Approximation Procedure
- Identify the target function and evaluation point.
- Select a known series or derive a Taylor series.
- State the center and convergence region.
- Transform the series if needed.
- Choose a truncation degree.
- Evaluate the polynomial efficiently.
- Apply a valid remainder estimate.
- Increase the degree or change strategy if the tolerance is not met.
- Round only after the error requirement is satisfied.
A Reliable Strategy When Convergence Is Slow
- Check the distance from the center.
- Look for a closer Taylor center.
- Factor out a known value if possible.
- Reduce the argument.
- Use a transformed standard series.
- Only then increase the number of terms if needed.
Worked Example: Approximate \(e^{0.1}\)
Use
\[ e^x = 1+x+\frac{x^2}{2!} +\frac{x^3}{3!}+\cdots. \]With the cubic truncation,
\[ e^{0.1} \approx 1+0.1+\frac{0.1^2}{2} +\frac{0.1^3}{6}. \]Thus
\[ \boxed{ e^{0.1}\approx1.1051667 }. \]Worked Example: Approximate \(\sin(0.2)\)
Use the cubic approximation
\[ \sin x\approx x-\frac{x^3}{6}. \]Therefore,
\[ \sin(0.2) \approx 0.2-\frac{0.2^3}{6}. \]Hence
\[ \boxed{ \sin(0.2)\approx0.1986667 }. \]Worked Example: Approximate \(\ln(1.1)\)
Here \(x=0.1\) in
\[ \ln(1+x) = x-\frac{x^2}{2} +\frac{x^3}{3} -\cdots. \]Using three terms,
\[ \ln(1.1) \approx 0.1-\frac{0.1^2}{2} +\frac{0.1^3}{3}. \]Thus
\[ \boxed{ \ln(1.1)\approx0.0953333 }. \]Worked Example: Approximate a Rational Function
Approximate
\[ \frac1{1-0.2} \]using the first four geometric terms:
\[ 1+0.2+0.2^2+0.2^3. \]Therefore,
\[ \boxed{ \frac1{0.8}\approx1.248 }. \]The exact value is \(1.25\), so the truncation is already close.
Worked Example: Approximate an Integral
Use
\[ e^{-x^2} \approx 1-x^2+\frac{x^4}{2} \]to approximate
\[ \int_0^{0.5}e^{-x^2}\,dx. \]Integrating the polynomial gives
\[ \boxed{ 0.5-\frac{0.5^3}{3} +\frac{0.5^5}{10} }. \]Practice 1: Exponential Approximation
Use the degree-\(2\) Maclaurin polynomial to approximate
\[ e^{0.05}. \]Practice 2: Sine Approximation
Use
\[ \sin x\approx x-\frac{x^3}{6} \]to approximate \(\sin(0.1)\).
Practice 3: Cosine Approximation
Use
\[ \cos x\approx1-\frac{x^2}{2} \]to approximate \(\cos(0.2)\).
Practice 4: Logarithmic Approximation
Use the first three nonzero terms of
\[ \ln(1+x) \]to approximate \(\ln(1.05)\).
Practice 5: Transform a Known Series
Write the first four nonzero terms of a power series for
\[ e^{-2x}. \]Practice 6: Leading Behavior
Use a Maclaurin series to identify the leading nonzero term of
\[ 1-\cos x \]as \(x\to0\).
Practice 7: Approximate an Integral
Using
\[ e^{-x^2} \approx 1-x^2, \]approximate
\[ \int_0^{0.2}e^{-x^2}\,dx. \]Practice 8: Horner Form
Rewrite
\[ 1+x+\frac{x^2}{2}+\frac{x^3}{6} \]in Horner form.
Practice 9: Choose the Better Center
To approximate \(e^{3.02}\), explain why a Taylor expansion centered at \(3\) is likely to be computationally better than a Maclaurin expansion.
Practice 10: Explain the Workflow
Explain why a function approximation should include both a convergence condition and an error estimate rather than only a truncated polynomial.
Practice Answers
Practice 1
Using
\[ e^x\approx1+x+\frac{x^2}{2}, \]we obtain
\[ e^{0.05} \approx 1+0.05+\frac{0.05^2}{2} = \boxed{1.05125}. \]Practice 2
\[ \sin(0.1) \approx 0.1-\frac{0.1^3}{6} = \boxed{0.0998333\ldots}. \]Practice 3
\[ \cos(0.2) \approx 1-\frac{0.2^2}{2} = \boxed{0.98}. \]Practice 4
Set \(x=0.05\):
\[ \ln(1.05) \approx 0.05-\frac{0.05^2}{2} +\frac{0.05^3}{3}. \]Therefore,
\[ \boxed{ \ln(1.05)\approx0.0487917 }. \]Practice 5
Substitute \(-2x\) into
\[ e^u = 1+u+\frac{u^2}{2!} +\frac{u^3}{3!}+\cdots. \]Thus
\[ \boxed{ e^{-2x} = 1-2x+2x^2-\frac43x^3+\cdots }. \]Practice 6
\[ 1-\cos x = \frac{x^2}{2} -\frac{x^4}{24} +\cdots. \]Therefore the leading nonzero term is
\[ \boxed{\frac{x^2}{2}}. \]Practice 7
\[ \int_0^{0.2}(1-x^2)\,dx = \left[x-\frac{x^3}{3}\right]_0^{0.2}. \]Therefore,
\[ \boxed{ \int_0^{0.2}e^{-x^2}\,dx \approx 0.2-\frac{0.2^3}{3} \approx0.1973333 }. \]Practice 8
\[ \boxed{ 1+x\left( 1+x\left( \frac12+\frac{x}{6} \right) \right) }. \]Practice 9
A Taylor series centered at \(3\) uses the small displacement
\[ x-3=0.02. \]Higher powers of \(0.02\) shrink rapidly. A Maclaurin series instead uses powers of \(3.02\), so substantially more terms may be needed for comparable accuracy.
Practice 10
The convergence condition tells us where the infinite series legitimately represents the target function. The error estimate tells us how accurately the chosen finite truncation approximates that function. Without both, the polynomial alone does not certify either validity or accuracy.
What to Remember
A convergent series representation
\[ f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n \]produces finite polynomial approximations
\[ \boxed{ P_N(x)=\sum_{n=0}^{N}c_n(x-a)^n }. \]- choose or derive a valid series representation;
- respect its interval or radius of convergence;
- choose a useful center;
- transform known standard series whenever efficient;
- truncate at a degree appropriate to the desired accuracy;
- use a remainder estimate to certify the approximation;
- evaluate the resulting polynomial efficiently.
Series approximation turns an infinite mathematical representation into a finite, controlled computational tool.
Where We Go Next
Series are one kind of infinite process, but infinite mathematical constructions can also be built through repeated multiplication, nesting, and continued operations.
The next lesson, Products and Continued Processes, broadens the study beyond additive series to infinite products, continued fractions, nested expressions, and other processes defined by repeated structure.