Geometric Series

Definition

A geometric series is an infinite series where each term is a constant multiple of the previous term. It is a type of infinite series that follows a specific pattern, allowing for the calculation of the sum of the series under certain conditions.

5 Must Know Facts For Your Next Test

1. The sum of a geometric series can be calculated using the formula $S = a / (1 - r)$, where $a$ is the first term and $r$ is the common ratio.
2. A geometric series converges if the absolute value of the common ratio $|r| < 1$, and diverges if $|r| \geq 1$.
3. Geometric series are often used to model real-world phenomena that exhibit exponential growth or decay, such as compound interest and radioactive decay.
4. The comparison test for convergence can be used to determine whether a geometric series converges or diverges by comparing it to a known convergent or divergent series.
5. Power series, which are a generalization of geometric series, are used to represent functions as infinite sums and are the foundation of calculus.

Review Questions

• Explain how the concept of a geometric series is related to the topic of infinite series.
  • Geometric series are a specific type of infinite series, where each term is a constant multiple of the previous term. The study of geometric series is an important part of the broader topic of infinite series, as it provides a way to determine the convergence or divergence of certain types of infinite series and to calculate the sum of convergent series. Understanding the properties and behavior of geometric series is crucial for the analysis of infinite series, which is a key concept in calculus.
• Describe how the comparison test for convergence can be used to determine the convergence or divergence of a geometric series.
  • The comparison test for convergence states that if the terms of one series are eventually less than or equal to the corresponding terms of a known convergent series, then the original series also converges. In the context of geometric series, this test can be used to determine convergence by comparing the common ratio $r$ of the geometric series to the common ratio of a known convergent geometric series. If $|r| < 1$, then the geometric series converges; if $|r| \geq 1$, then the series diverges. This allows for a straightforward way to classify the behavior of a geometric series without needing to calculate the full sum.
• Explain how the concept of a geometric series is related to the topic of power series and their use in representing functions.
  • Power series are a generalization of geometric series, where each term is a constant multiple of the previous term raised to a power. Just as geometric series are used to model exponential growth and decay, power series can be used to represent a wide variety of functions as infinite sums. The properties of geometric series, such as the formula for the sum of a convergent series and the conditions for convergence, are directly applicable to power series. Understanding geometric series is therefore crucial for the study of power series and their applications in calculus, where they are used to approximate and analyze functions through infinite series representations.

Related terms

Infinite Series: An infinite series is the sum of an infinite sequence of numbers or terms, where each term is a function of the index of the sequence.

Common Ratio: The common ratio of a geometric series is the constant ratio between consecutive terms in the series.

Convergence: Convergence in the context of a geometric series refers to the series approaching a finite sum as the number of terms increases, rather than diverging to infinity.