Conditionally Convergent Series

analysis

Definition

Conditionally Convergent Series

An infinite series ${\sum }_{n=0}^{\infty }{a}_n$ is called conditionally convergent if the series itself is convergent, but the series of absolute values ${\sum }_{n=0}^{\infty }|a_n|$ is divergent.

$$\sum a_n\text{ converges}\wedge \sum |a_n|\text{ diverges}$$

Characteristics

Unlike absolutely convergent series, the sum of a conditionally convergent series is sensitive to the order of its terms.

Riemann’s Rearrangement Theorem

The most striking property of conditional convergence is described by Riemann’s Rearrangement Theorem, which states that such a series can be rearranged to converge to any real number or even to diverge.

Visual Representation

The following diagram illustrates the partial sums of the alternating harmonic series $\sum \frac{(-1{)}^{n+1}}{n}$, which is the classic example of conditional convergence.

Examples

• Alternating Harmonic Series: ${\sum }_{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\cdots =\ln 2$.
• Dirichlet Series: Certain Dirichlet series on the boundary of their convergence half-plane.