Diverges

Definition

A series or sequence diverges if it does not converge to a finite limit. This means the terms do not approach a specific value as they progress to infinity.

5 Must Know Facts For Your Next Test

1. A series is said to diverge if the partial sums do not approach a finite limit.
2. An infinite sequence diverges if its terms do not tend towards a single number as the index increases.
3. A common test for divergence is the nth-term test, which states that if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges.
4. Geometric series with $|r| \geq 1$ will always diverge.
5. If a series alternates in sign but does not satisfy the conditions of the Alternating Series Test, it may still diverge.

Review Questions

• What condition must be met for an infinite sequence to be considered divergent?
• Explain how the nth-term test can be used to determine if a series diverges.
• Provide an example of a geometric series that diverges and explain why.

Related terms

Converges: A sequence or series converges if its terms approach a specific, finite value as they progress to infinity.

Partial Sums: The sum of the first n terms of a series. Used to analyze convergence or divergence.

$nth$-term Test: $nth$-term test states that if $\lim_{n \to \infty}a_n \neq0$, then $\sum{a_n}$ diverges.