Divergence test

Definition

The Divergence Test is a method used to determine whether a given series diverges. If the limit of the sequence's terms does not equal zero, the series diverges.

5 Must Know Facts For Your Next Test

1. The Divergence Test is applied by taking the limit of $a_n$ as $n$ approaches infinity.
2. If $\lim_{{n \to \infty}} a_n \neq 0$, then the series $\sum a_n$ diverges.
3. If $\lim_{{n \to \infty}} a_n = 0$, the Divergence Test is inconclusive and other tests must be used.
4. The Divergence Test cannot confirm convergence; it can only indicate divergence.
5. The test is often one of the first steps in analyzing a series for convergence or divergence.

Review Questions

• What does it mean if $\lim_{{n \to \infty}} a_n = L$ where $L \neq 0$?
• Why can't the Divergence Test confirm that a series converges?
• How do you apply the Divergence Test to the series $\sum_{n=1}^\infty (-1)^n$?

Related terms

Convergence:

A property of a series where its partial sums approach a finite limit as more terms are added.

Integral Test:

A method to determine if an infinite series converges or diverges by comparing it to an improper integral.

Harmonic Series:

$\sum_{n=1}^{\infty} \frac{1}{n}$, which is known to diverge despite its terms approaching zero.