Conditional convergence

Definition

Conditional convergence occurs when an infinite series converges, but it does not converge absolutely. This means the series converges only when the terms are taken in a specific order.

5 Must Know Facts For Your Next Test

1. The Alternating Series Test can be used to determine if a series is conditionally convergent.
2. A series that is conditionally convergent will not remain convergent if all terms are replaced with their absolute values.
3. Conditional convergence implies that the positive and negative terms of the series balance each other out to some extent.
4. A common example of a conditionally convergent series is the alternating harmonic series: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$$
5. If rearranged, a conditionally convergent series can be manipulated to converge to different limits or even diverge.

Review Questions

• What test would you use to determine if an alternating series is conditionally convergent?
• What distinguishes conditional convergence from absolute convergence?
• Give an example of a conditionally convergent series and explain why it meets the criteria.

Related terms

Absolute Convergence:

A series is absolutely convergent if the sum of the absolute values of its terms is finite, ensuring convergence regardless of term arrangement.

Alternating Series Test:

A test that determines whether an alternating series converges by checking if the absolute value of terms decreases monotonically and approaches zero.

Divergence:

Occurs when a series does not converge, meaning its partial sums do not approach a finite limit.