Sequences and Series — AI Study Guide

Mastering Sequences and Series

Sequences are ordered lists of numbers with a defined pattern. A sequence converges if its terms approach a finite limit as n → ∞; it diverges otherwise. Infinite series are sums of sequence terms: Σ_{n=1}^∞ a_n = a_1 + a_2 + a_3 + ... The sum exists only if the sequence of partial sums converges. The two key examples: geometric series Σar^{n-1} = a/(1-r) for |r| < 1 (converges) and diverges for |r| ≥ 1; p-series Σ1/n^p converges if p > 1 and diverges if p ≤ 1.

Convergence tests determine whether a series converges without finding the exact sum. Key tests: Divergence Test (if a_n does not approach 0, the series diverges — but the converse is false!), Integral Test (if a_n = f(n) where f is continuous, positive, and decreasing, then series converges iff ∫_1^∞ f(x)dx converges), Comparison Test (compare with a known series), Limit Comparison Test (if lim a_n/b_n = L > 0, both series have same convergence behavior), Ratio Test (if |a_{n+1}/a_n| → r < 1, series converges absolutely).

Power series represent functions as infinite polynomials: Σc_n(x-a)^n. Every power series has a radius of convergence R — it converges absolutely for |x - a| < R, diverges for |x - a| > R, and the endpoints require separate testing. Taylor series are power series centered at a with coefficients c_n = f^{(n)}(a)/n!: f(x) = Σ f^{(n)}(a)/n! (x-a)^n. Maclaurin series are Taylor series centered at 0.

Key Maclaurin series to memorize: e^x = Σx^n/n!, sin x = Σ(-1)^n x^{2n+1}/(2n+1)!, cos x = Σ(-1)^n x^{2n}/(2n)!, ln(1+x) = Σ(-1)^{n+1} x^n/n for |x| ≤ 1 (x≠-1), 1/(1-x) = Σx^n for |x| < 1 (geometric series). Taylor series allow approximation of function values, evaluation of limits, and solution of differential equations when exact formulas are unavailable.

Frequently Asked Questions: Sequences and Series

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers: a1, a2, a3, ... A series is the sum of the terms of a sequence: a1 + a2 + a3 + ... = Σa_n. A sequence converges if its terms approach a finite limit. A series converges if the sequence of its partial sums (S_n = a1 + a2 + ... + a_n) approaches a finite limit. A sequence can converge even if its corresponding series diverges (e.g., 1/n → 0 but Σ1/n = ∞, the harmonic series).

What is a Taylor series?

A Taylor series is an infinite power series representation of a function centered at x = a: f(x) = Σ_{n=0}^∞ f^{(n)}(a)/n! · (x-a)^n = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... The series equals f(x) within the radius of convergence (the series converges and equals f(x) there). Maclaurin series are Taylor series centered at a = 0. Taylor polynomials are truncations of Taylor series that approximate f(x) near a.

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