Unit 10 · Infinite Sequences and Series

Infinite series ○ — The sum of the terms of an infinite sequence, written as Σaₙ. A series may add up to a finite value or fail to settle on one.

nth partial sum ● (core concept) — S_n — the sum of the first n terms of a series. Convergence of a series is decided by what happens to its partial sums.

Convergence of an infinite series ● (core concept) — An infinite series converges to a real number S (has sum S) if and only if the limit of its sequence of partial sums exists and equals S. Otherwise the series diverges.

Geometric series ● (core concept) — A series with a constant ratio r between successive terms. It converges exactly when |r| < 1, with sum a/(1 − r) for Σₙ₌₀^∞ arⁿ, and diverges when |r| ≥ 1.

nth Term Test for Divergence ● (core concept) — If the limit of the terms aₙ is not zero, the series Σaₙ diverges. If the limit is zero, the test is inconclusive — the series may converge or diverge.

Integral test ● (core concept) — For a series Σaₙ with positive, decreasing terms coming from a continuous function (aₙ = f(n)), the series and the improper integral of f over [1, ∞) either both converge or both diverge.

Harmonic series ● (core concept) — The series Σ1/n. It is the classic example of a series that diverges even though its terms approach zero.

p-Series ● (core concept) — The series Σ1/nᵖ. It converges if and only if p > 1, and diverges for p ≤ 1; the harmonic series is the p = 1 case.

Comparison test ● (core concept) — For series with nonnegative terms: if 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges; if aₙ ≥ bₙ ≥ 0 and Σbₙ diverges, then Σaₙ diverges.

Limit comparison test ● (core concept) — For series with positive terms: if the limit of aₙ/bₙ is a finite positive number, then Σaₙ and Σbₙ either both converge or both diverge.

Alternating series ● (core concept) — A series whose terms alternate in sign, such as Σ(−1)ⁿbₙ with bₙ > 0. The alternating harmonic series is a convergent example.

Alternating Series Test ● (core concept) — An alternating series converges if its positive parts bₙ are decreasing and approach zero. Only the convergence — not the sum — is established by this test.

Ratio test ● (core concept) — Compute L = lim |aₙ₊₁/aₙ|. The series converges absolutely if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.

Absolute convergence ● (core concept) — Σaₙ converges absolutely when Σ|aₙ| converges. Absolute convergence implies ordinary convergence.

Conditional convergence ● (core concept) — A series that converges but does not converge absolutely. Unlike absolutely convergent series — whose terms can be rearranged without changing the sum — rearranging a conditionally convergent series can change its value.

Alternating series error bound ● (core concept) — For an alternating series that converges by the Alternating Series Test, the error from stopping at a partial sum is at most the magnitude of the first omitted term.

Taylor polynomial ● (core concept) — The degree-n polynomial P_n(x) = Σₖ₌₀ⁿ f⁽ᵏ⁾(a)/k! · (x − a)ᵏ, whose coefficients come from the function's derivatives at x = a. Higher-degree Taylor polynomials generally approximate f better near a; centered at a = 0 it is called a Maclaurin polynomial.

Lagrange error bound ● (core concept) — A bound on the error of a Taylor polynomial approximation: the remainder satisfies |R_n(x)| ≤ M|x − a|ⁿ⁺¹/(n+1)!, where M is a bound on |f⁽ⁿ⁺¹⁾| over the interval. When the Taylor series alternates, the alternating series error bound can be used instead.

Power series ○ — A series of the form Σₙ₌₀^∞ aₙ(x − r)ⁿ, where the aₙ are coefficients and r is the center. Power series can represent functions on an interval.

Radius of convergence ● (core concept) — The number R such that a power series converges for |x − r| < R and diverges for |x − r| > R (R may be 0 or infinite). It is found using the ratio test.

Interval of convergence ● (core concept) — The complete set of x-values for which a power series converges: the open interval given by the radius of convergence, plus any endpoints where the series converges. Each endpoint must be tested separately.

Term-by-term differentiation and integration ● (core concept) — Differentiating or integrating a power series one term at a time produces a new power series with the same radius of convergence, representing the derivative or antiderivative of the original function.

Taylor series ● (core concept) — The infinite series Σₙ₌₀^∞ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ. A Taylor polynomial is a partial sum of the Taylor series, and a power series with positive radius of convergence is the Taylor series of the function it converges to.

Maclaurin series ● (core concept) — A Taylor series centered at a = 0. The series for 1/(1 − x) is a geometric series, and the series for sin x, cos x, and eˣ are the foundation for constructing Maclaurin series of other functions by substitution and algebraic manipulation.

Representing functions as power series ● (core concept) — A power series for a given function can be derived from a known series by term-by-term differentiation or integration, substitution, algebraic processes, or properties of geometric series.