Sequences & Series — Contextual Knowledge Study Guide (Calc II)

These flashcards cover key terms and definitions related to sequences and series, helping students review critical concepts for their exam.

Sequence

A list of numbers written in a definite order, usually defined by a formula or recursively.

Series

The sum of the terms of a sequence, written as ∑an.

Partial Sum (Sn)

The sum of the first n terms of a series, represented as Sn = a1 + a2 + … + an.

Convergence (Sequence)

A sequence converges if the limit as n approaches infinity exists and is finite.

Divergence (Sequence)

A sequence diverges if the limit does not exist or is infinite.

Convergent Series

A series converges if the sequence of partial sums approaches a finite limit.

Divergent Series

A series diverges if the partial sums do not approach a finite limit.

Geometric Series

A series of the form ∑arn, which converges if |r| < 1 and diverges if |r| ≥ 1.

Harmonic Series

A series of the form ∑1/n, which is divergent despite its terms approaching 0.

nth-Term Test

If the limit of the nth term as n approaches infinity is not 0, the series diverges; if it equals 0, the test is inconclusive.

Integral Test

If an = f(n) where f is positive, continuous, and decreasing, then ∑an converges or diverges with ∫f(x) dx.

Comparison Test

Compares the terms of two series to determine convergence or divergence based on inequalities.

Limit Comparison Test

Computes the limit of the ratio of terms to compare series; if the limit is positive and finite, both series behave the same.

Ratio Test

Determines convergence or divergence of a series based on the limit of the ratio of consecutive terms.

Root Test

Determines convergence or divergence using the limit of the nth root of absolute terms.

Alternating Series

A series of the form ∑(-1)nan with an > 0.

Alternating Series Test (Leibniz Test)

An alternating series converges if the terms decrease and the limit of the terms approaches 0.

Absolute Convergence

A series is absolutely convergent if the series of the absolute values of its terms converges.

Conditional Convergence

A series converges but is not absolutely convergent.

Power Series

A series of the form ∑cn(x−a)n.

Radius of Convergence

The distance from a where the power series converges, requiring endpoint tests.

Taylor Series

A power series representation of a function using derivatives at a point.

Maclaurin Series

A Taylor series centered at a = 0.