Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Real Analysis → Sequences and Series ↓


Power Series


In mathematics, particularly the field of real analysis, a power series is an infinite series that has the form:

S(x) = a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + ... = ∑(a_n(x - c)^n)
S(x) = a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + ... = ∑(a_n(x - c)^n)

Where a_n represents the coefficients of the series, c is a constant, and x represents the variable. The series is centered at c.

Understanding the components

Coefficients (a_n): These are constant numbers that multiply each term of the series. They significantly affect the behavior of the series and its convergence or divergence.

Center (c): This is the value around which the series is expanded. If c = 0, this simplifies the series into a 'Maclaurin series'. More generally, c allows the series to represent the function more flexibly on different intervals.

Variable (x): This is the argument of the function that the series represents, described as a variable term.

Convergence of power series

The convergence of a power series depends on the value of x and the behavior of the coefficients relative to the center. Whether or not a power series converges—that is, whether it adds up to a finite number—is determined primarily by the ratio test, which calculates the radius of convergence.

Radius of convergence

The radius of convergence, R, is the distance from the center c within which the series converges. Mathematically, we often use the formula:

R = 1 / limsup |a_n|^(1/n)
R = 1 / limsup |a_n|^(1/n)

If R is finite, then the interval of convergence is (c - R, c + R). If R = 0, then the power series converges only at c. If R = ∞, then it converges for all x.

Example of power series

Let us consider a simple power series example:

S(x) = 1 + x + x^2 + x^3 + ... = ∑(x^n)
S(x) = 1 + x + x^2 + x^3 + ... = ∑(x^n)

This series is centered at c = 0 The coefficients a_n are all 1. This series represents a geometric series which converges when |x| < 1 and has the sum:

S(x) = 1 / (1 - x), |x| < 1
S(x) = 1 / (1 - x), |x| < 1

Visualization with examples

Let's visualize the convergence of a power series using a geometric progression example:

1 X x^2 x^3 s(x)

This simple diagram shows circles representing the terms in a power series. As more terms are added (more circles), the series converges to a specific value on the y axis, which is the sum S(x).

Term by term differentiation and integration

A special property of power series is that they can be differentiated and integrated term by term within the interval of convergence. This makes series quite convenient to manipulate, especially in calculus. For example:

Step by step differentiation

If:

S(x) = ∑(a_n(x - c)^n)
S(x) = ∑(a_n(x - c)^n)

So the derivative is:

S'(x) = ∑(n * a_n * (x - c)^(n-1))
S'(x) = ∑(n * a_n * (x - c)^(n-1))

Term by term integration

The indefinite integral of a power series is:

∫S(x) dx = ∑(a_n/(n+1) * (x - c)^(n+1)) + C
∫S(x) dx = ∑(a_n/(n+1) * (x - c)^(n+1)) + C

where C is the constant of integration.

Applications of power series

Power series are tools that allow mathematicians and scientists to approximate complex functions. They are used to provide solutions to differential equations, evaluate integrals, and optimize functions in engineering, physics, and economics.

Maclaurin and Taylor series

The Maclaurin series is a special type of power series with center 0. It is a special case of the Taylor series, where:

f(x) = ∑(f^n(0) / n! * x^n)
f(x) = ∑(f^n(0) / n! * x^n)

And the Taylor series can be represented as:

f(x) = ∑(f^n(c) / n! * (x - c)^n)
f(x) = ∑(f^n(c) / n! * (x - c)^n)

Here, f^n denotes n derivative of the function evaluated at the point c or 0 for the Maclaurin series. These expansions are important in finding values of a function that cannot otherwise be expressed easily.

Example: exponential function

The exponential function e^x can be expressed by its Maclaurin series as follows:

e^x = ∑(x^n / n!)
e^x = ∑(x^n / n!)

This series converges for all real values of x.

Conclusion

Power series provide a convenient and powerful way to express functions as infinite series expansions. The key lies in understanding the behavior and convergence of these series. By considering the coefficients of each term and the point about which the series is concentrated, we can use power series to approximate, differentiate, and integrate complex functions. Thus, they play an essential role in both theoretical and applied mathematics.


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Power Series

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