Taylor and Maclaurin Series

GRE Math Subject · Axiom Academy

LESSON Taylor and Maclaurin Series Power series approximations for functions Maclaurin Series (about x = 0): Key insight: These series allow us to represent functions as polynomials, useful for approximation and analysis. Common Maclaurin Series (Memorize These!) Radius and Interval of Convergence Radius of Convergence (R): The radius R is found using the ratio or root test. The series converges for |x - a| < R. Interval of Convergence: Apply the ratio test and check endpoints separately (they may or may not converge). Example: The series Σ (x^n / n!) for eˣ converges for all x (R = ∞). The series Σ (x^n / n) for ln(1+x) converges on (-1, 1] (R = 1, but check endpoints). Recognize when a series matches a known form Use substitution to adapt standard series (e.g., sin(2x) from sin(x)) Check convergence at boundaries for interval questions Use series to evaluate limits: lim(x→0) sin(x)/x = 1 from series

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