Definition and Formula
Concept
Taylor series: infinite sum of terms representing a function as a polynomial around point a. Expresses f(x) in terms of derivatives at a. Provides local approximation.
Mathematical Formula
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^nComponents
- f^{(n)}(a): nth derivative at point a
- n!: factorial of n
- (x - a)^n: power term centered at a
- n from 0 to infinity: infinite degree
Derivation of Taylor Series
Starting Point
Function assumed infinitely differentiable at a. Express f(x) as polynomial plus remainder.
Using Repeated Differentiation
Set polynomial P_n(x) with unknown coefficients. Differentiate n times at a to solve coefficients via derivatives.
Resulting Series
Equate coefficients: c_n = f^{(n)}(a) / n!. Leads to known formula.
P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^kConvergence and Radius of Convergence
Definition
Series convergence: infinite sum approaches function value as n → ∞. Not guaranteed everywhere.
Radius of Convergence
Radius R: interval |x - a| < R where series converges. Determined by ratio/root test on coefficients.
Absolute and Uniform Convergence
Absolute convergence: sum of absolute terms converges. Uniform convergence: series converges uniformly on interval.
| Test | Condition | Result |
|---|---|---|
| Ratio Test | lim |a_{n+1}/a_n| = L | R = 1/L (if L ≠ 0) |
| Root Test | lim sup |a_n|^{1/n} = L | R = 1/L (if L ≠ 0) |
Maclaurin Series
Definition
Special case of Taylor series centered at a = 0. Simplifies formula.
Formula
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^nExamples
Common expansions: e^x, sin x, cos x at 0.
Common Examples
Exponential Function e^x
e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \cdotsSine Function sin x
\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdotsCosine Function cos x
\cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots| Function | Taylor / Maclaurin Expansion |
|---|---|
| e^x | 1 + x + x²/2! + x³/3! + ... |
| sin x | x - x³/3! + x⁵/5! - ... |
| cos x | 1 - x²/2! + x⁴/4! - ... |
Applications in Calculus and Analysis
Function Approximation
Approximates complex functions by polynomials. Enables numerical methods and simplification.
Solving Differential Equations
Series solutions to ODEs and PDEs via power series expansions.
Computing Limits and Derivatives
Facilitates limit calculation using series expansions. Derivative evaluation via term-wise differentiation.
Error Estimation and Remainder Term
Remainder Term Concept
Difference between function and nth partial sum. Measures approximation accuracy.
Lagrange Form of Remainder
R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}, \quad c \in (a, x)Bounding the Error
Use maximum of |f^{(n+1)}(t)| on interval to bound |R_n(x)|. Guides number of terms needed.
Operations on Taylor Series
Addition and Subtraction
Term-wise addition/subtraction of series with same center.
Multiplication
Cauchy product: multiply series coefficients via convolution.
Division and Composition
Division: find series for quotient via recursive formulas. Composition: substitute one series into another.
Taylor Series for Multivariable Functions
Extension to Several Variables
Expansion about point a = (a_1, ..., a_n) using partial derivatives.
Formula
f(\mathbf{x}) = \sum_{|\alpha| \ge 0} \frac{D^\alpha f(\mathbf{a})}{\alpha!} (\mathbf{x} - \mathbf{a})^\alphaMulti-index Notation
α = (α_1, ..., α_n), |α| = sum α_i, D^α partial derivatives, α! factorial product.
Limitations and Pitfalls
Non-analytic Functions
Functions not analytic at a: Taylor series may not converge to function.
Radius of Convergence Restrictions
Series valid only inside radius R; outside may diverge or misrepresent function.
Misuse in Numerical Computations
Truncation errors, floating-point limits. Requires error control for reliability.
Historical Background
James Gregory and Brook Taylor
Gregory (1671): early polynomial approximations. Taylor (1715): formalized series expansion named after him.
Development of Analysis
Central to calculus, analysis, numerical methods development in 18th-19th centuries.
Modern Usage
Foundation of perturbation theory, asymptotic expansions, computational algorithms.
References
- Rudin, W. Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976, pp. 195-210.
- Apostol, T. M. Mathematical Analysis, 2nd ed., Addison-Wesley, 1974, pp. 154-167.
- Stein, E. M., Shakarchi, R. Real Analysis: Measure Theory, Integration, & Hilbert Spaces, Princeton, 2005, pp. 301-315.
- Boas, M. L. Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005, pp. 120-135.
- Courant, R., John, F. Introduction to Calculus and Analysis, Vol. 1, Springer, 1999, pp. 220-240.