Divergent Sequence

Definition

A divergent sequence is a mathematical sequence where the terms move further and further apart, increasing without bound. This is in contrast to a convergent sequence, where the terms get closer and closer together, approaching a specific value.

5 Must Know Facts For Your Next Test

1. In a divergent geometric sequence, the common ratio is greater than 1, causing the terms to increase without bound.
2. The formula for the $n^{th}$ term of a divergent geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
3. Divergent sequences have no limit, as the terms continue to grow larger without approaching a specific value.
4. The sum of the terms in a divergent geometric sequence does not converge, and the series is said to be divergent.
5. Divergent sequences are often used to model exponential growth, such as compound interest or population growth.

Review Questions

• Explain the relationship between the common ratio and the divergence of a geometric sequence.
  • In a divergent geometric sequence, the common ratio must be greater than 1. This means that each term is multiplied by a value larger than 1, causing the terms to increase without bound. The larger the common ratio, the faster the sequence will diverge, with the terms growing exponentially larger. The common ratio is the key factor that determines whether a geometric sequence will be divergent or convergent.
• Describe how the formula for the $n^{th}$ term of a divergent geometric sequence differs from that of a convergent geometric sequence.
  • The formula for the $n^{th}$ term of a divergent geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio. This formula is the same as for a convergent geometric sequence, but the key difference is that in a divergent sequence, the common ratio $r$ is greater than 1. In a convergent sequence, the common ratio is less than 1. This distinction in the common ratio is what determines whether the sequence will diverge or converge as the terms are calculated.
• Analyze how the behavior of a divergent geometric sequence differs from that of a convergent geometric sequence in terms of the sum of the series.
  • The fundamental difference between divergent and convergent geometric sequences lies in the behavior of their series. For a divergent geometric sequence, the sum of the terms does not converge to a finite value. Instead, the series continues to grow larger without bound. This is because the common ratio is greater than 1, causing the terms to increase exponentially. In contrast, for a convergent geometric sequence, the sum of the terms does converge to a finite value. This is because the common ratio is less than 1, causing the terms to decrease in magnitude and approach a specific limit. The divergent nature of the series is a key characteristic that distinguishes divergent geometric sequences from their convergent counterparts.

Related terms

Geometric Sequence: A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a common ratio.

Common Ratio: The common ratio in a geometric sequence is the constant value by which each term is multiplied to get the next term.

Convergent Sequence: A convergent sequence is a sequence where the terms get closer and closer together, approaching a specific value.