Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis → Differentiation ↓


Taylor Series


Introduction

The Taylor series is a very powerful tool in mathematics used to approximate more complex functions with simpler polynomial expressions. The series provides a way of representing functions as infinite sums calculated from the values of their derivatives at a point. Essentially, it breaks down a complex curve into the sum of its tangent (linear), quadratic, cubic, and higher-order terms.

Understanding the basics

Let's start by understanding the basic form of the Taylor series. For a function ( f ) that is infinitely differentiable at a point ( a ), the Taylor series of ( f ) at ( a ) is given by:

f(x) = f(a) + f'(a)(x - a) + frac{f''(a)}{2!}(x - a)^2 + frac{f'''(a)}{3!}(x - a)^3 + cdots

This formula represents an infinite sum where ( n! ) (n factorial) is the product of all positive integers up to ( n ), and the ( n )-th derivative of ( f ) is evaluated at the point ( a ).

Example of Taylor series

Let's take a simple function like ( exp(x) = e^x ) and look at its Taylor series at ( a = 0 ). This series is often called the Maclaurin series, which is a special case of the Taylor series where ( a = 0 ).

e^x = 1 + x + frac{x^2}{2!} + frac{x^3}{3!} + frac{x^4}{4!} + cdots

The Maclaurin series for ( e^x ) is derived from its derivatives, which are all ( e^x ). Evaluated at ( x = 0 ), all derivatives are equal to 1, giving the series:

e^0 = 1, quad (e^x)'|_{x=0} = 1, quad (e^x)''|_{x=0} = 1, quad ldots

Thus, each term is simply ( frac{x^n}{n!} ).

Visualizing the Taylor series


  
    
    

    0

    
    
    Sine function: sin(x)

    
    
    Taylor approximation up to degree 3

This SVG example compares the sine function to its third-degree Taylor approximation. Notice how close they are near ( x = 0 ), which shows the power of the Taylor series in function approximation.

Construction of higher order Taylor series

Now, let's build a more detailed example - the Taylor series of the natural logarithm function, ( ln(1 + x) ), at ( a = 0 ).

Calculate the derivatives:

f(x) = ln(1 + x) f'(x) = frac{1}{1 + x} f''(x) = -frac{1}{(1 + x)^2} f'''(x) = frac{2}{(1 + x)^3} ldots

Evaluating at ( x = 0 ) gives: 

f(0) = 0, quad f'(0) = 1, quad f''(0) = -1, quad f'''(0) = 2, quad ldots

Thus, the Taylor series is:

ln(1 + x) = x - frac{x^2}{2} + frac{x^3}{3} - frac{x^4}{4} + cdots

This series converges for ( -1 < x leq 1 ). The radius of convergence is the distance between the center of expansion ( a ) and the point where the series loses its validity, which is 1 in this case.

Convergence of Taylor series

An essential aspect of Taylor series is to understand the concept of convergence. Not all Taylor series converge to the function they are supposed to represent. The interval in which the Taylor series converges to its function is called the "interval of convergence."

For example, consider the geometric series:

frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + cdots

This series converges to ( frac{1}{1-x} ) only for ( |x| < 1 ). If you choose any value of ( x ) with an absolute value greater than or equal to 1, the series sum no longer approaches the actual function value.

Applications of Taylor series

Taylor series are used in a variety of areas of science and engineering:

  1. Approximation: Complex functions can be approximated by using few polynomial terms, by truncating the series.
  2. Numerical analysis: Many numerical algorithms depend on approximating functions with polynomials.
  3. Physics: Used to solve problems related to oscillations and wave equations in quantum mechanics and other branches.
  4. Engineering: Simplifies complex models for control systems and signal processing tasks.

Taylor series and error analysis

An important part of using Taylor series is understanding the error associated with truncating the series. The remaining term ( R_n(x) ) provides an estimate of how accurate a truncated series approximation is.

The remainder of the ( n ) -th order Taylor polynomial is given by:

R_n(x) = frac{f^{(n+1)}(c)}{(n+1)!}(xa)^{n+1}

where ( c ) is a value in the interval between ( a ) and ( x ). This term helps to determine how close the estimate is to the true value of the function.

Conclusion

Taylor series are a cornerstone of calculus and analysis, providing essential methods for approximating functions, analyzing convergence, and applying results in a wide variety of scientific fields. Its utility reaches far beyond basic mathematics, underpinning breakthroughs in computational methods, physics, engineering, and beyond.


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Taylor Series

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