Divergent Series

Definition

A divergent series is a series whose partial sums do not converge to a finite value. In other words, the sum of the terms in the series continues to increase without bound, rather than approaching a specific number. This behavior is in contrast to a convergent series, where the partial sums do approach a finite value.

5 Must Know Facts For Your Next Test

1. A series is divergent if the absolute value of the terms does not approach 0 as the number of terms increases.
2. The harmonic series is a classic example of a divergent series, where the partial sums grow without bound.
3. Divergent series have no well-defined sum, unlike convergent series, which have a finite sum.
4. Determining whether a series is convergent or divergent is an important step in analyzing the behavior of the series.
5. Divergent series have many applications in mathematics, such as in the study of infinite products and power series.

Review Questions

• Explain the difference between a convergent and a divergent series, and provide an example of each.
  • A convergent series is one where the partial sums approach a finite value as the number of terms increases, whereas a divergent series is one where the partial sums continue to grow without bound. A classic example of a convergent series is the geometric series $\sum_{n=0}^\infty ar^n$, where $|r| < 1$. An example of a divergent series is the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$, where the partial sums grow without limit.
• Describe the role of partial sums in determining whether a series is convergent or divergent.
  • Partial sums play a crucial role in determining the convergence or divergence of a series. For a convergent series, the partial sums approach a finite value as the number of terms increases. Conversely, for a divergent series, the partial sums continue to grow without bound. By analyzing the behavior of the partial sums, one can classify a series as either convergent or divergent, which is an important step in understanding the series' behavior and potential applications.
• Explain how the concept of divergent series is related to the study of infinite products and power series in mathematics.
  • Divergent series have important connections to the study of infinite products and power series in mathematics. Infinite products, which are the product of an infinite number of factors, can sometimes be expressed in terms of divergent series, allowing for their analysis and manipulation. Additionally, the study of power series, which are series of the form $\sum_{n=0}^\infty a_nx^n$, often involves determining whether the series converges or diverges for different values of the variable $x$. Understanding the properties of divergent series is crucial in this context, as it helps to delineate the regions of convergence and divergence for power series, which have numerous applications in mathematics and physics.

Related terms

Convergent Series: A convergent series is a series whose partial sums approach a finite value as the number of terms increases.

Partial Sums: The partial sums of a series are the sums of the first $n$ terms of the series, where $n$ is a positive integer.

Harmonic Series: The harmonic series is a divergent series, where the terms are given by the reciprocals of the positive integers, $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$.