Sequences and Series Cheat Sheet

A set of vocabulary flashcards summarizing key concepts related to sequences and series in calculus.

Sequence

An ordered list of numbers, usually defined as {a_n}, where n is a positive integer.

Convergence

A sequence {an} converges to L if \lim{n\to\infty} a_n = L.

Divergence

A sequence diverges if \lim{n\to\infty} an does not exist.

Monotonic Sequence Theorem

If a sequence is both monotonic (increasing or decreasing) and bounded, it converges.

Squeeze Theorem

If an ≤ bn ≤ cn and \lim an = \lim cn = L, then \lim bn = L.

Geometric Sequence

A sequence defined by a_n = ar^n; converges if |r| < 1, diverges otherwise.

Harmonic Sequence

A sequence defined by a_n = 1/n; converges to 0.

Factorial Sequence

A sequence defined by a_n = n!; diverges to infinity.

Series

The sum of the terms of a sequence: Sn = a1 + a2 + … + an.

Partial Sums

Defined as Sn = \sum{k=1}^{n} a_k.

Geometric Series

A series of the form \sum_{n=0}^{\infty} ar^n; converges if |r| < 1.

p-Series

A series of the form \sum \frac{1}{n^p}; converges if p > 1.

nth-Term Test for Divergence

If \lim{n\to\infty} an ≠ 0, then \sum a_n diverges.

Integral Test

If f(n) is positive, continuous, and decreasing, then \sum an converges if \int1^\infty f(x)dx converges.

Comparison Test

If 0 ≤ an ≤ bn and \sum bn converges, then \sum an converges.

Limit Comparison Test

If \lim{n\to\infty} \frac{an}{bn} = c where c > 0, then \sum an and \sum b_n behave the same.

Alternating Series Test

An alternating series converges if an is decreasing and \lim{n\to\infty} a_n = 0.

Ratio Test

If \lim{n\to\infty} \left| \frac{a{n+1}}{a_n} \right| = r: converges if r < 1, diverges if r > 1.

Root Test

If \lim{n\to\infty} \sqrt[n]{|an|} = L: converges if L < 1, diverges if L > 1.

Power Series

A series of the form \sum{n=0}^{\infty} cn (x - a)^n.

Radius of Convergence

Determined using the Ratio Test; defines the interval where the series converges.

Taylor Series

A series representing a function as \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n.

Maclaurin Series Expansions

Special case of Taylor series at a = 0 for functions like e^x, sin x, cos x.

Summary of Strategies

Use different tests for convergence based on series types; prioritize tests based on series properties.