Power Series

Definition and Basic Concepts

Power Series Form

General form: sum from n=0 to infinity of a_n(x - c)ⁿ. Variable: x. Center: c. Coefficients: a_n real or complex numbers.

Infinite Polynomial

Infinite sum resembling polynomial but with infinite terms. Terms are powers of (x - c) scaled by coefficients.

Domain Concept

Defined where series converges. Convergence determines domain of validity.

Examples of Power Series

Geometric series: sum of xⁿ. Exponential series: sum of xⁿ/n!. Sine and cosine expansions as power series.

Convergence of Power Series

Pointwise Convergence

Series converges at point x if partial sums approach finite limit as n → ∞.

Uniform Convergence

Convergence on interval where partial sums uniformly approximate function within error tolerance.

Absolute Convergence

Sum of absolute values converges. Implies convergence of original series.

Conditional Convergence

Series converges but does not converge absolutely. Rare in power series context.

Radius and Interval of Convergence

Radius of Convergence

Distance from center c within which series converges. Determined by limit superior or root/ratio tests.

Calculation Methods

Ratio test: R = 1 / limsup |a_n+1/a_n|. Root test: R = 1 / limsup |a_n|^1/n.

Interval of Convergence

Real interval (c - R, c + R) where series converges. May include or exclude endpoints depending on tests.

Behavior at Endpoints

Convergence must be checked separately at x = c ± R. Can be convergent or divergent.

Coefficients and Function Representation

Role of Coefficients

Each a_n scales nth power term. Determines shape and behavior of represented function.

Uniqueness

Coefficients uniquely define analytic function within radius of convergence.

Recovering Coefficients

From function f(x), coefficients via derivatives at center c: a_n = f⁽ⁿ⁾(c)/n!.

Formal Power Series vs. Analytic Functions

Formal series: algebraic objects ignoring convergence. Analytic functions: series with positive radius converging to function.

an = \frac{f^{(n)}(c)}{n!}

Operations on Power Series

Addition and Subtraction

Termwise addition/subtraction: sum of series coefficients a_n ± b_n. Radius at least minimum of both.

Multiplication

Cauchy product: coefficients c_n = sum from k=0 to n of a_kb_n-k. Radius ≥ minimum radii.

Differentiation

Termwise differentiation valid within radius. New series sum n a_n(x - c)^n-1. Same radius.

Integration

Termwise integration valid within radius. New series sum a_n(x - c)ⁿ⁺¹/(n+1). Same radius.

Given: f(x) = Σ aₙ(x - c)ⁿDerivative: f'(x) = Σ n aₙ (x - c)ⁿ⁻¹Integral: ∫f(x)dx = C + Σ aₙ (x - c)ⁿ⁺¹ / (n + 1) 

Taylor and Maclaurin Series

Taylor Series Definition

Expansion of function f(x) about point c into power series using derivatives at c.

Maclaurin Series

Special case of Taylor series about c = 0.

Formula

f(x) = sum n=0 to ∞ [f⁽ⁿ⁾(c)/n!] (x - c)ⁿ.

Convergence Conditions

Function must be infinitely differentiable at c. Convergence radius depends on function behavior.

f(x) = Σ (f⁽ⁿ⁾(c) / n!) (x - c)ⁿ, n=0 to ∞

Analytic Functions and Power Series

Definition of Analyticity

Function is analytic at c if equals its power series expansion in neighborhood of c.

Relationship to Differentiability

Analytic implies infinitely differentiable. Converse not always true.

Examples of Analytic Functions

Exponential, sine, cosine, rational functions with no singularities at c.

Non-analytic but Infinitely Differentiable

Functions like e^-1/x² at 0 are smooth but not analytic there.

Examples of Power Series

Geometric Series

Sum of xⁿ, n=0 to ∞, converges for |x| < 1. Sum = 1/(1-x).

Exponential Function

eˣ = sum of xⁿ/n!, converges ∀ x ∈ ℝ.

Sine and Cosine Series

sin x = sum (-1)ⁿ x^(2n+1)/(2n+1)!. cos x = sum (-1)ⁿ x^(2n)/(2n)!.

Logarithmic Series

ln(1+x) = sum (-1)ⁿ⁺¹ xⁿ/n, for -1 < x ≤ 1.

Geometric: Σ xⁿ = 1/(1 - x), |x| < 1Exponential: Σ xⁿ/n! = eˣ, ∀ xSine: Σ (-1)ⁿ x^(2n+1)/(2n+1)!Cosine: Σ (-1)ⁿ x^(2n)/(2n)!Logarithm: Σ (-1)ⁿ⁺¹ xⁿ / n, -1 < x ≤ 1 

Applications in Calculus

Function Approximation

Use partial sums to approximate functions near center with known error bounds.

Solving Differential Equations

Power series methods solve ODEs when closed forms unavailable.

Evaluating Limits and Integrals

Series expansions facilitate limit evaluation and integral approximation.

Numerical Analysis

Basis for numerical methods, including spectral methods and series truncations.

Common Tests for Convergence

Ratio Test

Evaluate limit of |a_n+1/a_n|. If <1, series converges absolutely.

Root Test

Evaluate limit of |a_n|^1/n. Same criteria as ratio test.

Comparison Test

Compare with known convergent/divergent series for boundary points.

Endpoint Testing

Check convergence at interval boundaries individually.

Limitations and Pitfalls

Radius of Convergence Limits

Series represents function only within radius; outside, may diverge or misrepresent.

Non-analytic Functions

Functions not analytic cannot be represented by power series.

Endpoint Ambiguity

Convergence at endpoints requires careful separate analysis.

Computational Issues

Truncation errors and slow convergence for some functions.

Summary and Key Points

Power Series Definition

Infinite sum of coefficients times powers of (x - c).

Convergence Domain

Determined by radius and interval of convergence.

Taylor and Maclaurin Series

Express analytic functions as power series via derivatives.

Operations Validity

Addition, multiplication, differentiation, integration valid within radius.

Applications

Approximation, solving equations, numerical methods.

References

• Rudin, W. "Principles of Mathematical Analysis." McGraw-Hill, 3rd ed., 1976, pp. 150-170.
• Apostol, T. M. "Mathematical Analysis." Addison-Wesley, 2nd ed., 1974, pp. 300-320.
• Stein, E. M., and Shakarchi, R. "Real Analysis: Measure Theory, Integration, and Hilbert Spaces." Princeton University Press, 2005, pp. 200-225.
• Bronshtein, I. N., Semendyayev, K. A. "Handbook of Mathematics." Springer-Verlag, 5th ed., 2007, pp. 100-110.
• Pugh, C. C. "Real Mathematical Analysis." Springer, 2nd ed., 2015, pp. 215-235.