Divergent Series

Definition

A divergent series is a mathematical series where the sum of the terms does not converge to a finite value. In other words, as more terms are added, the sum of the series continues to increase without bound, rather than approaching a specific value.

5 Must Know Facts For Your Next Test

1. Divergent series have no finite sum, meaning the series continues to grow larger without bound as more terms are added.
2. The harmonic series, where the terms are the reciprocals of the positive integers, is a classic example of a divergent series.
3. Whether a series is convergent or divergent depends on the behavior of the terms in the series as more are added.
4. Geometric series can be either convergent or divergent, depending on the value of the common ratio between terms.
5. Divergent series have important applications in mathematics, physics, and computer science, despite their lack of a finite sum.

Review Questions

• Explain the key difference between a convergent and a divergent series.
  • The key difference between a convergent and a divergent series is the behavior of the sum as more terms are added. In a convergent series, the sum of the terms approaches a finite value, whereas in a divergent series, the sum continues to increase without bound. Convergent series have a well-defined sum, while divergent series do not have a finite sum that can be calculated.
• Describe the relationship between the harmonic series and divergent series.
  • The harmonic series, where the terms are the reciprocals of the positive integers, is a classic example of a divergent series. As more terms are added to the harmonic series, the sum continues to grow larger without approaching a finite value. This divergent behavior of the harmonic series is an important result in mathematical analysis and has applications in various fields, such as the study of infinite sums and the foundations of calculus.
• Analyze the role of the common ratio in determining whether a geometric series is convergent or divergent.
  • The common ratio in a geometric series plays a crucial role in determining whether the series is convergent or divergent. If the absolute value of the common ratio is less than 1, the series is convergent, and the sum of the series approaches a finite value. However, if the absolute value of the common ratio is greater than or equal to 1, the series is divergent, and the sum of the series continues to grow without bound. This property of geometric series is a fundamental result in the study of infinite series and has important applications in various areas of mathematics and science.

Related terms

Convergent Series: A convergent series is a mathematical series where the sum of the terms approaches a finite value as more terms are added.

Harmonic Series: The harmonic series is a divergent series where the terms are the reciprocals of the positive integers.

Geometric Series: A geometric series is a series where each term is a constant multiple of the previous term, and the series can be either convergent or divergent depending on the value of the common ratio.