zuai-logo

Glossary

A

Alternating Harmonic Series

Criticality: 2

A series similar to the harmonic series but with alternating signs, typically represented as $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$.

Example:

Unlike the regular harmonic series, the alternating harmonic series 1−1/2+1/3−1/4+...1 - 1/2 + 1/3 - 1/4 + ... actually converges to a finite value.

C

Converges

Criticality: 3

A series *converges* if its sequence of partial sums approaches a finite, specific limit. This means the sum of its terms adds up to a definite number.

Example:

If you keep adding terms of ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}, the sum will eventually approach a finite value, meaning the series converges.

D

Diverges

Criticality: 3

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

Example:

Even though the terms get smaller, the harmonic series diverges because its sum continues to grow without bound.

H

Harmonic Series

Criticality: 3

A specific type of p-series where the exponent p equals 1, represented as the sum of the reciprocals of the positive integers.

Example:

The series 1 + 1/2 + 1/3 + 1/4 + ... is a classic example of a harmonic series and is known to diverge.

S

Summation Notation

Criticality: 2

A concise mathematical notation using the Greek capital letter sigma ($\Sigma$) to represent the sum of a sequence of terms.

Example:

The expression ∑n=14(2n)\sum_{n=1}^{4} (2n) uses summation notation to represent the sum 2(1)+2(2)+2(3)+2(4)2(1) + 2(2) + 2(3) + 2(4).

p

p-series

Criticality: 3

A series of the form $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$, where p is a positive real number. Its convergence depends on the value of p.

Example:

The series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^{2}} is a p-series with p=2, which converges because p > 1.