Conditional convergence is convergence with a condition that is a series is said to be conditionally convergent if it converges, but not absolutely. This means that while the series ∑a_n​ converges, the series of the absolute values ∑∣a_n∣ diverges.

A classic example of a conditionally convergent series is the alternating harmonic series ∑(−1)ⁿ⁺¹1/n, which converges to ln⁡(2), despite the fact that the harmonic series ∑1/n diverges. This divergence of the series of absolute values indicates that the positive and negative terms in the original series play a crucial role in its convergence.

In this article, we will discuss the concept of Conditional Convergence in detail including examples and various test to find weather series is conditionally convergent or not.

What is Conditional Convergence?

Conditional convergence is a term used in mathematics to describe a specific behavior of an infinite series and refers to a property of an infinite series in mathematics where the series converges, but the series formed by taking the absolute values of its terms diverges.

To be more specific, any conditionally convergent series converges to a finite limit only because of the specific arrangement and cancellation of its terms, despite the fact that the magnitude of the terms grows large without bound.

Definition of Conditional Convergence

A series [Tex]\sum_{n=0}^{\infty} a_n[/Tex]​ is said to be conditionally convergent if:

• The series [Tex]\sum_{n=0}^{\infty} a_n[/Tex]an​ converges to a finite value.
• The series of the absolute values of its terms,[Tex]\sum_{n=0}^{\infty} |a_n|[/Tex], diverges.

Examples of Conditionally Convergent Series

Some of the common examples of series that converges conditionally are:

Alternating Harmonic Series

The alternating harmonic series is a well-known example of a conditionally convergent series:

[Tex]\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} = 1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \frac{1}{5} – \cdots[/Tex]

This series converges to ln⁡(2), but the harmonic series [Tex]\sum_{n=1}^{\infty} \frac{1}{n}[/Tex], which consists of the absolute values of the terms, diverges​​​​.

Alternating Series with Logarithmic Terms

Another example is the series involving logarithmic terms:

[Tex]\sum_{n=1}^{\infty} (-1)^n \frac{\ln(n)}{n}[/Tex]

This series converges conditionally. However, the series of absolute values [Tex] \sum_{n=1}^{\infty} \frac{\ln(n)}{n}​ [/Tex] diverges​​.

Series Involving Trigonometric Functions

Series that include alternating trigonometric functions can also be conditionally convergent, such as:

[Tex]\sum_{n=1}^{\infty} (-1)^n \frac{\cos(n)}{n}[/Tex]

This series converges conditionally, but the series [Tex]\sum_{n=1}^{\infty} \frac{\cos(n)}{n}[/Tex] does not converge absolutely​​.

Alternating p-Series

For 0 < p ≤ 1, the alternating p-series:

[Tex]\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n^p}[/Tex] is conditionally convergent.

For example, when p = 1, it reduces to the alternating harmonic series, which is conditionally convergent. For p < 1, the series still converges conditionally because the positive p-series [Tex]\sum_{n=1}^{\infty} \frac{1}{n^p}​[/Tex] diverges

Tests for Conditional Convergence

Some of the most common test for conditional convergence are:

• Alternating Series Test
• Dirichlet’s Test
• Abel’s Test

Alternating Series Test

The Alternating Series Test is used specifically for alternating series of the form [Tex]\sum (-1)^n a_n[/Tex], where a_n​ is a positive, decreasing sequence that approaches zero.

Conditions for the test:

• a_n​ > 0
• a_n+1​ ≤ a_n​ (the terms a_n​​ are monotonically decreasing)
• [Tex]\lim_{n \to \infty} a_n = 0[/Tex]

If these conditions are met, the series ∑(−1)nan\sum (-1)^n a_n∑(−1)nan​ converges.

Dirichlet’s Test

Dirichlet’s Test can be applied to a series of the form [Tex]\sum a_n b_n[/Tex], where:

• a_n​​ is a sequence of complex numbers such that ∑a_n​​ is bounded.
• b_n​ is a monotonically decreasing sequence of positive real numbers tending to zero.

If these conditions are satisfied, the series ∑a_n​b_n converges.

Abel’s Test

Abel’s Test is another method for determining the conditional convergence of series of the form ∑a_n​b_n​​, where:

• a_n​ is a sequence whose partial sums are bounded.
• b_n​​ is a monotonically decreasing sequence that converges to zero.

If these conditions are met, the series ∑a_n​b_n​​ converges.

Absolute vs Conditional Convergence

The key difference between absolute and conditional convergence are listed in the following table: