Divergent Series

A divergent series is a series in mathematics that does not have a finite sum. In other words, the sum of the terms in a divergent series does not converge to a specific value but instead grows without bound.

Characteristics of Divergent Series

There are several ways to identify a divergent series:

• The partial sums of a divergent series do not approach a finite number.
• The terms of a divergent series do not approach zero.
• When the series is broken down into smaller sums, each of these sums diverges.

Examples of Divergent Series

One famous example of a divergent series is the harmonic series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

This series diverges, as the sum of its terms grows without bound as more terms are added.

Types of Divergent Series

There are different types of divergent series, such as:

• Conditionally convergent series: A series that is convergent when the terms are rearranged in a specific order, but diverges in its original form.
• Sequences with no limit: Series where the terms oscillate between positive and negative values such that no finite sum exists.
• Series diverging to infinity: Series where the terms grow without bound, leading to an infinite sum.