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Glossary

A

Absolute Convergence

Criticality: 3

A series $\sum a_n$ is absolutely convergent if the series formed by taking the absolute value of each term, $\sum |a_n|$, converges.

Example:

The series ∑n=1∞(−1)nn2\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} is absolutely convergent because ∑n=1∞∣(−1)nn2∣=∑n=1∞1n2\sum_{n=1}^{\infty} \left|\frac{(-1)^n}{n^2}\right| = \sum_{n=1}^{\infty} \frac{1}{n^2} converges (it's a p-series with p=2>1p=2 > 1).

Alternating Series Test

Criticality: 3

A test used to determine the convergence of an alternating series, which requires the terms to be positive, decreasing in magnitude, and approach zero as n approaches infinity.

Example:

To show that ∑n=1∞(−1)n+1ln⁡(n+1)\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\ln(n+1)} converges, you would apply the Alternating Series Test, checking that 1ln⁡(n+1)\frac{1}{\ln(n+1)} is positive, decreasing, and approaches zero.

C

Conditional Convergence

Criticality: 3

A series $\sum a_n$ is conditionally convergent if the series itself converges, but the series of its absolute values, $\sum |a_n|$, diverges.

Example:

The alternating harmonic series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} is conditionally convergent because it converges by the Alternating Series Test, but its absolute value, ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n} (the harmonic series), diverges.

D

Direct Comparison Test

Criticality: 2

A test for convergence or divergence of a series by comparing its terms to those of another series whose convergence or divergence is already known.

Example:

To determine if ∑n=1∞1n2+1\sum_{n=1}^{\infty} \frac{1}{n^2+1} converges, you could use the Direct Comparison Test by comparing it to the convergent p-series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}.

Divergent

Criticality: 2

A series is divergent if its sequence of partial sums does not approach a finite limit, meaning the sum grows infinitely large or oscillates without settling.

Example:

The series ∑n=1∞n\sum_{n=1}^{\infty} n is divergent because its terms continuously increase, causing the sum to grow without bound.

H

Harmonic Series

Criticality: 2

The series $\sum_{n=1}^{\infty} \frac{1}{n}$, which is a specific type of p-series where $p=1$.

Example:

When evaluating the absolute convergence of the alternating harmonic series, you'll encounter the harmonic series ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}, which is a classic example of a divergent series.

P

P-series

Criticality: 2

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which converges if $p > 1$ and diverges if $p \leq 1$.

Example:

The series ∑n=1∞1n4\sum_{n=1}^{\infty} \frac{1}{n^4} is a p-series with p=4p=4, which immediately tells us it converges.