Divergence of a series

Definition

The divergence of a series occurs when the sum of its terms does not approach a finite limit as more terms are added. A divergent series either increases without bound, decreases without bound, or oscillates indefinitely.

5 Must Know Facts For Your Next Test

1. A series is divergent if the sequence of its partial sums does not converge to a finite limit.
2. If the $n$-th term of a series does not approach zero as $n$ approaches infinity, the series is divergent.
3. The comparison test can be used to determine if a series diverges by comparing it to another known divergent series.
4. The ratio test and root test are common methods for testing the divergence of a series.
5. A geometric series with a common ratio $|r| \geq 1$ diverges.

Review Questions

Related terms

Convergence of a Series: Occurs when the sum of all terms in an infinite series approaches a specific finite value as more terms are added.

$n$-th Term Test: A test stating that if $\lim_{{n \to \infty}} a_n \neq 0$, then the series $\sum_{n=1}^{\infty}a_n$ diverges.

Partial Sum:

The sum of the first $n$ terms in a sequence, often denoted as $S_n$. Used to analyze convergence or divergence.