Study Guide for Series and Convergence Tests

What is a sequence in mathematics?

An ordered list of numbers, typically defined by a formula or recurrence relation.

What is a series in mathematics?

The sum of the terms of a sequence.

What do Partial Sums refer to?

The sum of the first n terms, used to analyze convergence.

What conditions must a function f(x) meet to apply the Integral Test?

The function must be positive, continuous, and decreasing.

How does the Integral Test help with series?

It determines if a corresponding series behaves like an improper integral.

What is the formula for the surface area of a function rotated about the x-axis?

A = ∫ab 2πy sqrt(1 + (dy/dx)²) dx.

What does a recurrence relation express?

A term using previous terms.

What is an example of a recurrence relation?

an = r * an−1.

What distinguishes closed-form formulas from recurrence relations?

A closed-form formula provides an explicit expression for any term without recursion.

What is a monotonic sequence?

A sequence that is either increasing or decreasing.

What is the difference between absolute and conditional convergence?

Absolute convergence means the series converges absolutely, while conditional convergence means the series converges but not absolutely.

When is the Ratio Test particularly useful?

For series involving factorial terms (n!) or exponential terms (r^n).

What form does a geometric series take?

∑ ar^n.

What is the purpose of the Integral Evaluation for series tests?

To compute improper integrals to analyze convergence.

Which test should you use if terms have factorials or exponentials?

Try the Ratio or Root Test.

What does the Limit Comparison Test do?

It compares a given series with a known reference series.