Taylor Series: Complete Guide, Formula, Calculator, Examples, Error Bounds, and Exam Practice
A Taylor series rewrites a function as an infinite polynomial built from the function’s derivatives at a chosen center. It is one of the most powerful ideas in calculus because it connects local behavior, approximation, infinite series, error estimation, and real-world modelling.
This page explains Taylor polynomials, Maclaurin series, common expansions, convergence, Lagrange remainder, exam-style scoring guidance, course relevance, and step-by-step examples. Use the interactive calculator below to approximate \(e^x\), \(\sin x\), \(\cos x\), \(\ln x\), and \(\frac{1}{x}\) using Taylor polynomials.
Taylor Polynomial Calculator
Choose a function, center \(a\), degree \(n\), and input value \(x\). The tool builds a Taylor polynomial around \(a\), estimates \(f(x)\), compares it with the actual value, and shows the absolute error.
Lagrange Error Bound Helper
If \(\left|f^{(n+1)}(t)\right|\le M\) between \(a\) and \(x\), the Lagrange remainder satisfies:
\[ |R_n(x)| \le \frac{M|x-a|^{n+1}}{(n+1)!} \]
What Is a Taylor Series?
A Taylor series is an infinite polynomial representation of a function near a chosen point. If a function \(f(x)\) has enough derivatives at a point \(a\), then the Taylor series centered at \(a\) is:
\[ f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots \]
\[ f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^k \]
The expression looks advanced, but the idea is direct. A Taylor series uses the value of the function, the slope, the curvature, and higher derivative information at one point to build a polynomial that imitates the function. The more terms you include, the more information the polynomial uses. In many situations, adding more terms improves the approximation near the center \(a\).
A finite version is called a Taylor polynomial. The degree-\(n\) Taylor polynomial is:
\[ P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k \]
The polynomial \(P_n(x)\) is not usually equal to \(f(x)\) everywhere. It is an approximation. The difference between the actual value and the approximation is called the remainder or error:
\[ R_n(x)=f(x)-P_n(x) \]
Maclaurin Series: Taylor Series Centered at Zero
A Maclaurin series is a Taylor series centered at \(a=0\). It is not a different concept; it is a special case of Taylor series. When \(a=0\), the formula becomes:
\[ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots \]
\[ f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k \]
Maclaurin series are common in school and university calculus because many famous functions have clean derivative patterns at zero. For example, \(e^x\) keeps the same derivative forever. The functions \(\sin x\) and \(\cos x\) cycle through four derivative patterns. The geometric series gives a foundation for \(\frac{1}{1-x}\), and from it students can derive related expansions such as \(\ln(1+x)\) and \(\arctan x\).
Standard Taylor and Maclaurin Series Table
The following formulas are essential. Students should know the formula, the interval of convergence, and how the expansion can be adapted by substitution, differentiation, or integration.
| Function | Series Expansion | Common Interval / Condition | Important Use |
|---|---|---|---|
| \(e^x\) | \(\displaystyle e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\) | All real \(x\) | Growth models, differential equations, approximations |
| \(\sin x\) | \(\displaystyle \sin x=\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{(2k+1)!}\) | All real \(x\) | Small-angle approximations, waves, oscillations |
| \(\cos x\) | \(\displaystyle \cos x=\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k}}{(2k)!}\) | All real \(x\) | Physics, geometry, harmonic motion |
| \(\frac{1}{1-x}\) | \(\displaystyle \frac{1}{1-x}=\sum_{k=0}^{\infty}x^k=1+x+x^2+x^3+\cdots\) | \(|x|<1\) | Geometric series, rational functions |
| \(\ln(1+x)\) | \(\displaystyle \ln(1+x)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{x^k}{k}\) | \(-1| Logarithm estimates and alternating series | |
| \(\arctan x\) | \(\displaystyle \arctan x=\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{2k+1}\) | \(-1\le x\le1\) | Inverse tangent estimates and \(\pi\) approximations |
| \((1+x)^r\) | \(\displaystyle (1+x)^r=1+rx+\frac{r(r-1)}{2!}x^2+\frac{r(r-1)(r-2)}{3!}x^3+\cdots\) | Usually \(|x|<1\) for non-integer \(r\) | Binomial approximations and fractional powers |
Diagram: Function vs Taylor Polynomial
The diagram below shows the main idea visually. A Taylor polynomial matches a function at a center point. A first-degree polynomial matches the value and slope. A second-degree polynomial also matches curvature. Higher-degree polynomials can capture more local shape.
Why Taylor Series Matter
1. Approximation
Many functions are difficult to calculate directly. A Taylor polynomial can approximate values using addition, multiplication, powers, and factorials. This is why Taylor series are used in numerical methods and scientific computing.
2. Local Behaviour
Taylor polynomials reveal how a function behaves near a point. The constant term gives the value, the linear term gives slope, the quadratic term gives curvature, and higher terms refine the local shape.
3. Error Control
Taylor theory does not only approximate. It also helps estimate how wrong the approximation might be. Error bounds are essential in exams, engineering, physics, and numerical analysis.
4. Function Representation
Some functions can be represented as infinite power series. This allows students to transform, differentiate, integrate, and analyze functions using algebraic series tools.
How to Build a Taylor Polynomial Step by Step
To build a Taylor polynomial, you need a function \(f(x)\), a center \(a\), and a degree \(n\). The process is systematic. First, calculate the function value at \(a\). Next, calculate the first derivative, second derivative, third derivative, and so on until you reach the \(n\)-th derivative. Then substitute those derivative values into the Taylor polynomial formula.
- Choose the center: Decide the point \(a\) where the polynomial should match the function.
- Find derivatives: Compute \(f(a), f'(a), f''(a), f^{(3)}(a), \ldots, f^{(n)}(a)\).
- Divide by factorials: Each \(k\)-th derivative value is divided by \(k!\).
- Use powers of \((x-a)\): The \(k\)-th term uses \((x-a)^k\).
- Write the polynomial: Add terms from \(k=0\) to \(k=n\).
- Check accuracy: Use the remainder formula, a graph, or direct comparison when possible.
Example 1: Maclaurin Series for \(e^x\)
The function \(f(x)=e^x\) is the cleanest Taylor series example because every derivative is still \(e^x\). At \(x=0\), every derivative equals \(1\). Therefore:
\[ f(0)=1,\quad f'(0)=1,\quad f''(0)=1,\quad f^{(3)}(0)=1 \]
Substitute these values into the Maclaurin formula:
\[ e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots \]
A fourth-degree Maclaurin polynomial is:
\[ P_4(x)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!} \]
If \(x=1\), then:
\[ P_4(1)=1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24} \]
\[ P_4(1)=2.708333\ldots \]
The actual value is \(e\approx2.7182818\), so the fourth-degree polynomial is already close. More terms make the approximation stronger.
Example 2: Maclaurin Series for \(\sin x\)
For \(f(x)=\sin x\), the derivatives cycle:
\[ \sin x,\quad \cos x,\quad -\sin x,\quad -\cos x,\quad \sin x,\ldots \]
At \(x=0\), the values are:
\[ \sin 0=0,\quad \cos 0=1,\quad -\sin 0=0,\quad -\cos 0=-1 \]
That is why the Maclaurin series for \(\sin x\) contains only odd powers:
\[ \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots \]
The small-angle approximation \(\sin x\approx x\) comes from the first nonzero term of this series. It is useful only when \(x\) is measured in radians and is close to zero. A better approximation is:
\[ \sin x\approx x-\frac{x^3}{6} \]
Example 3: Maclaurin Series for \(\cos x\)
The derivative pattern for \(\cos x\) also cycles. At zero, the nonzero terms occur for even powers:
\[ \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots \]
The simplest approximation is \(\cos x\approx1\) for very small \(x\). A stronger approximation is:
\[ \cos x\approx1-\frac{x^2}{2} \]
This approximation appears in physics, especially when simplifying pendulum motion, wave behaviour, and small-angle models. In exams, students must remember that trigonometric Taylor series use radians. If degrees are used accidentally, the numerical approximation becomes incorrect.
Example 4: Taylor Polynomial for \(\ln x\) Around \(a=1\)
The natural logarithm is often expanded around \(a=1\) because \(\ln 1=0\). Let \(f(x)=\ln x\). The derivatives are:
\[ f'(x)=\frac{1}{x},\quad f''(x)=-\frac{1}{x^2},\quad f^{(3)}(x)=\frac{2}{x^3},\quad f^{(4)}(x)=-\frac{6}{x^4} \]
At \(a=1\), these become:
\[ f(1)=0,\quad f'(1)=1,\quad f''(1)=-1,\quad f^{(3)}(1)=2,\quad f^{(4)}(1)=-6 \]
The Taylor series is:
\[ \ln x=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+\cdots \]
This is equivalent to the famous formula:
\[ \ln(1+u)=u-\frac{u^2}{2}+\frac{u^3}{3}-\frac{u^4}{4}+\cdots \]
where \(u=x-1\). The condition \(|u|<1\) means this expansion works best when \(x\) is near \(1\).
Example 5: Taylor Series for \(\frac{1}{x}\) Around \(a=1\)
The function \(\frac{1}{x}\) can be expanded around \(a=1\). Write:
\[ \frac{1}{x}=\frac{1}{1+(x-1)} \]
Using the geometric series:
\[ \frac{1}{1+u}=1-u+u^2-u^3+u^4-\cdots \]
Substitute \(u=x-1\):
\[ \frac{1}{x}=1-(x-1)+(x-1)^2-(x-1)^3+(x-1)^4-\cdots \]
This is valid when \(|x-1|<1\), which means:
\[
0
A Taylor polynomial is useful only when you understand its error. The error after degree \(n\) is: \[
R_n(x)=f(x)-P_n(x)
\]
One of the most common exam tools is the Lagrange remainder: \[
R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}
\]
for some value \(c\) between \(a\) and \(x\). Since \(c\) is usually unknown, students often use a maximum bound: \[
|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}
\]
where \(M\) is a maximum value of \(\left|f^{(n+1)}(t)\right|\) on the interval between \(a\) and \(x\).
This formula is heavily tested because it connects derivatives, intervals, factorials, powers, and approximation accuracy.
Some Taylor series are alternating series. If the terms decrease in magnitude and approach zero, then the alternating
series error is no larger than the first omitted term: \[
|R_n|\le |a_{n+1}|
\]
For example: \[
\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots
\]
If you approximate \(\sin x\) using terms up to \(x^5\), the error can often be bounded by the magnitude of the next
term: \[
\left|\frac{x^7}{7!}\right|
\]
This method is common for \(\sin x\), \(\cos x\), \(\ln(1+x)\), and \(\arctan x\), but only when the alternating series
conditions are satisfied.
A Taylor series is an infinite power series. That means convergence matters. Some series converge for every real number,
while others converge only within a limited interval. The radius of convergence tells you how far from the center the
series can be trusted as an infinite representation.
For a power series centered at \(a\): \[
\sum_{k=0}^{\infty}c_k(x-a)^k
\]
the radius of convergence \(R\) means the series converges when: \[
|x-a|
The endpoints \(x=a-R\) and \(x=a+R\) must be checked separately. A common method is the ratio test: \[
\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right|<1
\]
If the limit produces an inequality involving \(x\), solve it to find the radius. For example, the geometric series: \[
\sum_{k=0}^{\infty}x^k
\]
converges when: \[
|x|<1
\]
so its radius of convergence is \(R=1\). At \(x=1\), the series becomes \(1+1+1+\cdots\), which diverges. At \(x=-1\),
it becomes \(1-1+1-1+\cdots\), which does not converge in the usual sense.
Taylor series appear most directly in advanced calculus courses. They are especially important in AP Calculus BC,
IB Mathematics: Analysis and Approaches HL, university calculus, engineering mathematics, physics, numerical methods,
and mathematical modelling.
Taylor series is not a standalone exam. It is assessed inside broader calculus courses. The most direct school-level
exam connection is AP Calculus BC, where Taylor and Maclaurin series belong to the infinite sequences and series unit.
In IB, Taylor-style thinking is most relevant to higher-level calculus and approximation work.
Taylor series questions are not scored separately as a public final score. They contribute to the overall exam score
inside a course. For AP Calculus BC, Taylor series can appear in multiple-choice or free-response tasks. A strong answer
normally shows correct series setup, correct coefficients, correct interval or radius of convergence when required,
and a valid justification for any approximation or error bound.
Use this self-check table to estimate your readiness for Taylor series questions. This is not an official exam conversion.
It is a practical study guide for students.
A strong Taylor series revision plan should not begin with memorizing formulas only. First, review derivatives until the
patterns feel natural. Next, practice factorials and summation notation. Then learn the main Maclaurin series table.
After that, practice substitutions such as replacing \(x\) by \(2x\), \(x^2\), or \(-x\). Once substitutions feel easy,
move to radius and interval of convergence. Finally, practice error estimation with both Lagrange remainder and alternating
series bounds.
A focused seven-day plan can work well. On day one, revise derivatives and factorials. On day two, learn \(e^x\),
\(\sin x\), and \(\cos x\). On day three, learn geometric, logarithmic, arctangent, and binomial expansions.
On day four, practice building Taylor polynomials from derivatives. On day five, practice convergence tests. On day six,
practice error bounds. On day seven, complete mixed exam-style questions under timed conditions and review every mistake. A Taylor series represents a function as an infinite sum of terms built from its derivatives at a chosen center \(a\). The Taylor series formula is \(\displaystyle f(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^k\). A Maclaurin series is a Taylor series centered at \(a=0\), so the terms use powers of \(x\). A Taylor series is infinite. A Taylor polynomial uses only finitely many terms and gives an approximation. Derivatives describe local behavior. The first derivative gives slope, the second derivative gives curvature, and higher derivatives refine the function’s local shape. The Lagrange error bound estimates the maximum possible error after using a Taylor polynomial of degree \(n\): \(\displaystyle |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}\). Yes. Taylor and Maclaurin series are part of the infinite sequences and series material in AP Calculus BC. Yes, trigonometric Taylor series such as \(\sin x\) and \(\cos x\) require \(x\) to be measured in radians. No. Some functions have Taylor series that converge only on certain intervals, and some functions do not equal their Taylor series everywhere. Master derivative patterns, memorize the standard expansions, practice substitutions, check convergence carefully, and always justify approximation error when required.
Taylor series turn calculus into a powerful approximation system. They allow functions to be represented by polynomials,
help estimate values that are difficult to calculate directly, and provide a method for understanding local behavior.
The core formula is simple once the structure is understood: take derivatives at a center, divide by factorials, and
multiply by powers of \((x-a)\). The challenge is not only building the series, but also knowing when it converges,
how accurate it is, and how to justify the error.
For students preparing for advanced mathematics exams, Taylor series should be studied as a complete skill set:
formulas, derivations, convergence, approximation, error bounds, and communication. If you can explain why a polynomial
approximates a function, how the coefficients are created, and how large the error might be, then you are no longer
memorizing Taylor series. You are using them as a real calculus tool.Taylor Remainder and Error Bounds
Alternating Series Error Bound
Radius and Interval of Convergence
Taylor Series in AP, IB, and Advanced Calculus Courses
Course / Exam Where Taylor Series Fit What Students Should Master AP Calculus BC Infinite sequences and series; Taylor and Maclaurin series; error bounds; radius and interval of convergence Build series, identify convergence, use Taylor polynomials, estimate error, connect series to functions IB Math AA HL Higher-level calculus and series concepts; Taylor approximations may appear in advanced problem-solving contexts Understand derivatives, approximations, proof-style reasoning, and calculator/non-calculator interpretation A-Level Further Mathematics Maclaurin expansions, binomial expansions, approximation, and calculus applications Use standard expansions, substitute into series, and simplify expressions accurately University Calculus Power series, convergence, Taylor theorem, analytic functions, numerical approximation Prove results, bound errors, derive expansions, and apply series to modelling Engineering / Physics Linearization, small-angle approximations, numerical simulation, perturbation methods Choose appropriate terms, control error, and interpret approximation limits Latest Exam Timetable Snapshot for Taylor-Series-Relevant Courses
Exam 2026 Timetable Information Format / Notes AP Calculus BC Monday, May 11, 2026, 8 AM local time Hybrid digital exam. Multiple-choice section is 45 questions in 1 hour 45 minutes and counts for 50%. Free-response section is 6 questions in 1 hour 30 minutes and counts for 50%. IB Mathematics: Analysis and Approaches HL Paper 1: Thursday, May 14, 2026 afternoon. Paper 2: Friday, May 15, 2026 morning. Paper 3: Wednesday, May 20, 2026 afternoon. Paper durations vary by level. HL Mathematics paper 3 is 1 hour 15 minutes in the May 2026 schedule. IB Mathematics: Applications and Interpretation HL Paper 1: Thursday, May 14, 2026 afternoon. Paper 2: Friday, May 15, 2026 morning. Paper 3: Wednesday, May 20, 2026 afternoon. Useful for modelling and approximation contexts, although exact Taylor-series coverage depends on course pathway and teacher plan. Score Guidelines and Score Table
AP Score General Meaning Taylor Series Skill Level Usually Needed 5 Extremely well qualified Can derive series, use convergence tests, manage endpoints, justify error bounds, and handle unfamiliar series transformations. 4 Very well qualified Can solve most Taylor polynomial and Maclaurin series problems with minor errors in notation or endpoint details. 3 Qualified Understands standard series and basic Taylor polynomial construction but may struggle with harder convergence and remainder arguments. 2 Possibly qualified Recognizes some formulas but has inconsistent accuracy with coefficients, factorials, signs, or intervals. 1 No recommendation Needs to rebuild prerequisite skills: derivatives, factorials, power rules, series notation, and algebraic simplification. IB Grade General Meaning Taylor-Series-Relevant Evidence 7 Excellent performance Clear reasoning, accurate algebra, strong calculus foundations, correct interpretation, and precise communication. 6 Very strong performance Mostly correct method with strong fluency across derivatives, approximations, and problem interpretation. 5 Good performance Good conceptual understanding, but may lose marks on complex manipulation or justification details. 4 Satisfactory performance Handles routine calculus but needs more confidence with series notation and advanced reasoning. 3 and below Limited to weak performance Requires focused revision on derivatives, algebra, functions, and structured problem-solving. RevisionTown Taylor Series Mastery Score
Mastery Score Level What You Can Do Next Step 90–100% Exam-ready advanced Derive and apply Taylor series, prove convergence, estimate error, and solve unfamiliar problems. Practice timed free-response questions and mixed convergence tasks. 75–89% Strong Use common expansions and build Taylor polynomials accurately, with occasional endpoint or error-bound mistakes. Review Lagrange remainder and interval of convergence cases. 60–74% Developing Recognize formulas and solve routine problems, but may confuse signs, factorials, or degree terms. Create a formula sheet and complete 20 targeted practice questions. 40–59% Foundation level Understands the purpose of Taylor series but needs help with derivatives and notation. Review derivatives, factorials, summation notation, and polynomial basics. Below 40% Start again carefully Needs prerequisite rebuilding before attempting exam-level Taylor series questions. Study functions, derivatives, and basic sequences first. Common Mistakes in Taylor Series
Mistake Why It Happens Correct Approach Forgetting factorials Students write derivative values but forget the \(k!\) denominator. Every Taylor term has \(\frac{f^{(k)}(a)}{k!}\). Using \(x^k\) instead of \((x-a)^k\) Confusing Taylor and Maclaurin series. Use \(x^k\) only when the center is \(a=0\). Wrong signs in sine and cosine Missing the derivative cycle. Write the derivative pattern before building the series. Ignoring convergence Treating every power series as valid for all \(x\). Find the radius and check endpoints when asked. Incorrect error bound Using the wrong derivative order. For degree \(n\), the Lagrange bound uses the \((n+1)\)-th derivative. Using degrees instead of radians Trigonometric calculators often show degree mode. Taylor series for \(\sin x\) and \(\cos x\) require radians. Practice Questions
Answers
Study Plan for Taylor Series
Frequently Asked Questions
What is a Taylor series?
What is the Taylor series formula?
What is a Maclaurin series?
What is the difference between a Taylor series and a Taylor polynomial?
Why do Taylor series use derivatives?
What is the Lagrange error bound?
Are Taylor series tested in AP Calculus BC?
Do Taylor series require radians?
Does every function equal its Taylor series?
How can I get better at Taylor series?
Conclusion
