Properties of Power Series

Learning Objectives

• Combine power series by addition or subtraction.
• Create new power series by multiplying by a variable's power or via substitution.
• Multiply two power series together to form a single combined series.
• Differentiate and integrate power series term-by-term.

Key Terms & Glossary

• Power Series: An infinite series of the form $\sum c_n(x-a)^n$ representing a function as a polynomial.
• Interval of Convergence: The continuous range of $x$-values for which a power series resolves to a finite value.
• Term-by-Term Differentiation: Deriving an entire power series by applying the derivative rule to each individual term.
• Term-by-Term Integration: Integrating an entire power series by applying the antiderivative rule to each individual term.

The "Big Idea"

Power series behave almost exactly like infinitely long polynomials. Rather than computing difficult Taylor series from scratch, you can take a known simple series (like $\frac{1}{1-x}$) and algebraically combine, substitute, differentiate, or integrate it to discover the series for complex or non-elementary functions. This enables us to solve differential equations that lack elementary solutions!

Formula / Concept Box

Hierarchical Outline

• Combining Power Series
  • Addition and Subtraction: Combine coefficients of matching terms when intervals of convergence overlap.
  • Multiplication by a Power: Shift the series index naturally by distributing $x^m$.
  • Substitution: Replace the primary variable $x$ with $cx^m$ to find series for composite functions.
• Calculus of Power Series
  • Term-by-Term Differentiation: Moving the derivative operator directly inside the summation symbol.
  • Term-by-Term Integration: Moving the integral operator directly inside the summation symbol.
  • Evaluating the Boundaries: Radius remains consistent, but interval endpoint convergence must be manually checked.

Visual Anchors

Definition-Example Pairs

• Term-by-Term Differentiation

  • Definition: Differentiating an infinite series one polynomial term at a time.
  • Real-world Example: In signal processing, finding the velocity of an alternating current wave by deriving its positional power series term-by-term.

• Interval of Convergence

  • Definition: The continuous band of $x$-values where the series accurately represents the function.
  • Real-world Example: A physical calculator evaluating $\sin(x)$ uses a power series that is only programmed to operate accurately within a specific convergence interval to save memory.

Comparison Tables

Worked Examples

[!NOTE] Constructing a Series via Algebraic Substitution Find the power series and the interval of convergence for the function $f(x) = \frac{x^3}{1-x^2}$.

▶Click to expand step-by-step solution

1. Identify a related known series: We know the fundamental geometric series: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$.
2. Apply substitution: Replace $x$ with $x^2$. $\frac{1}{1-x^2} = \sum_{n=0}^{\infty} (x^2)^n = \sum_{n=0}^{\infty} x^{2n}$
3. Multiply by the remaining variable ($x^3$): $\frac{x^3}{1-x^2} = x^3 \sum_{n=0}^{\infty} x^{2n} = \sum_{n=0}^{\infty} x^{2n+3}$
4. Determine the interval of convergence: The original series converges for $|x| < 1. Our substitution requires |x^2| < 1, which mathematically simplifies to |x| < 1$. Thus, the interval of convergence is $(-1, 1)$.

Muddy Points & Cross-Refs

[!WARNING] Endpoint Behavior is Tricky! A major "muddy point" is assuming the exact interval of convergence never changes during calculus operations. While the overall radius ($R) stays identical, convergence at the exact boundaries (x = -R$ and $x = R$) can "break" during differentiation or "heal" during integration. You must manually test the endpoints using limit tests after differentiating or integrating!

Checkpoint Questions

1. If you memorized the geometric series for $\frac{1}{1-x}, what two algebraic steps would you take to find the series for \frac{x^2}{1+x}$?
2. What happens to the radius of convergence when you integrate a power series term-by-term?
3. Why does the starting index of a summation often shift from $n=0$ to $n=1$ when differentiating a power series?
4. True or False: If a series converges at $x=1, its term-by-term derivative is guaranteed to converge at x=1$ as well.