Maclaurin Series

Special case of Taylor series centered at a = 0 (simpler to calculate)

Expanded Taylor Series

Explicit form showing first few terms with derivatives and factorials

Ratio Test Formula

Most common method: take the limit of the ratio of consecutive coefficients

Convergence Rules

• If |x - c| < R: Series converges
• If |x - c| > R: Series diverges
• If |x - c| = R: Test separately

Special Cases

• R = 0: Converges only at center
• R = ∞: Converges everywhere
• 0 < R < ∞: Finite radius

What is a power series?

A power series is an infinite series of the form ∑(n=0 to ∞) aₙ(x - c)ⁿ, where aₙ are coefficients and c is the center. It represents a function as an infinite sum of powers of (x - c). Power series are fundamental in calculus and mathematical analysis for function approximation and solving differential equations.

What is the difference between Taylor and Maclaurin series?

A Taylor series is a power series expansion of a function centered at any point a: ∑[f⁽ⁿ⁾(a)/n!](x - a)ⁿ. A Maclaurin series is a special case of Taylor series centered at a = 0: ∑[f⁽ⁿ⁾(0)/n!]xⁿ. Maclaurin series are simpler to calculate but only approximate functions around x = 0, while Taylor series can be centered anywhere.

How do you find the power series of a function?

To find a power series: 1) Calculate derivatives f'(a), f''(a), f'''(a)... at center point a, 2) Apply Taylor series formula: f(x) = ∑[f⁽ⁿ⁾(a)/n!](x - a)ⁿ, 3) Substitute derivative values: f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ..., 4) Simplify and identify the pattern, 5) Write the series using sigma notation.

What is radius of convergence?

The radius of convergence R is the distance from the center point within which the power series converges to the function. Use the ratio test: R = lim(n→∞)|aₙ/aₙ₊₁|. The series converges for |x - c| < R, diverges for |x - c| > R, and requires separate testing at the boundary |x - c| = R. Some series have R = ∞ (converge everywhere) or R = 0 (converge only at center).

What are common power series expansions?

Common power series include: eˣ = ∑(xⁿ/n!) = 1 + x + x²/2! + x³/3! + ..., sin(x) = ∑((-1)ⁿx^(2n+1)/(2n+1)!) = x - x³/3! + x⁵/5! - ..., cos(x) = ∑((-1)ⁿx^(2n)/(2n)!) = 1 - x²/2! + x⁴/4! - ..., ln(1+x) = ∑((-1)^(n+1)xⁿ/n) = x - x²/2 + x³/3 - ..., and 1/(1-x) = ∑xⁿ = 1 + x + x² + x³ + .... These are fundamental in calculus.

How accurate is a power series approximation?

Accuracy depends on three factors: 1) Number of terms - more terms give better approximation, 2) Distance from center - accuracy decreases as you move away from the center point, 3) Radius of convergence - series is only valid within this radius. The remainder term (error) can be estimated using Taylor's Remainder Theorem. Near the center, even a few terms provide excellent accuracy.

Why do we use power series?

Power series are used because they: 1) Approximate complicated functions with simple polynomials for easier calculation, 2) Solve differential equations that lack closed-form solutions, 3) Enable numerical computations in calculators and computers (e.g., computing sin, cos, eˣ), 4) Analyze function behavior near specific points, 5) Provide rigorous error bounds, and 6) Allow integration and differentiation term-by-term within the radius of convergence.