Diverges

Definition

A series diverges if the sum of its terms does not approach a finite number as more terms are added. Divergence implies that the series either increases without bound, decreases without bound, or oscillates indefinitely.

5 Must Know Facts For Your Next Test

1. A series is said to diverge if its partial sums do not converge to a specific value.
2. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ is a classic example of a divergent series.
3. If the absolute value of terms in an infinite series does not approach zero, the series diverges.
4. The comparison test can be used to determine divergence by comparing with a known divergent series.
5. The ratio test and root test are common methods for determining whether a series converges or diverges.

Review Questions

• What does it mean for a series to diverge?
• Give an example of a divergent series and explain why it diverges.
• Which tests can be used to determine if a series diverges?

Related terms

Convergent Series: A series whose partial sums approach a specific finite limit as the number of terms grows.

Partial Sum: The sum of the first $n$ terms in a sequence, often used to analyze convergence or divergence.

$p$-Series: $p$-series is given by $\sum_{n=1}^{\infty} \frac{1}{n^p}$ and converges if $p > 1$; otherwise, it diverges.